I Wave Scintillations in the Ionosphere

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Radio Wave Scintillations in the Ionosphere





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 formulation of the problem. Based on measurements from


radar, and propagation experhnents, ionospheric irregularities

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

While earlier measurements indicated a spectral index of about 4, there is recent evidence showing that 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 signal, scintillation index, r m s phase fluctua- tions, correlation functions, power spectra, distriiutions, etc., are investipted. Effects of multiple scattering are


Expedmental results concerning irregularity structures and signal statics are presented.

These results are compared with theoretical predictions. 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 function and the temporal moments.

Results including numerical simulations are discussed Finally, some future efforts in ionospheric scintillation studies in the areas of transion- aspheric communication and space and geophysics are recommended.



A . History of Ionosphere 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 correlation between fluctuations recorded at two 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 earth’s atmosphere. Indeed, as later observations confirmed [4 ] -[ l o ] , this marked the f i i t observation of the ionosphere scintillation phenomenon.

After the f i t artificial satellite was launched in 1957, it became possible to observe ionosphere scintillations using radio transmissions from the satellite [ 1 1


-[ 151


The interest in the study of this phenomenon has continued in the last two decades. In general, the interests are twofold. On the one hand, the study of the scintillation problem is directly related t o the transionospheric communication problems such 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 Manuscript received September 18, 1981; revised January 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 the 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. In the early seventies, the discovery of scintillation at gigahertz frequencies [ 171, [ 181 presented an additional challenge to the field. Two satellite beacon experiments specially designed for scintillation studies, the ATS-6 and the Wideband Satellite [ 191, [ 201, have provided us with new observational data that helped to enhance o m knowledge of the scintillation phenomenon. These include coherent multiple frequency data for both amplitude 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 of the irregularities.

On the theoretical side, ionospheric scintillation was first studied in terms of the thin phase screen theory [ 221, [ 231.

Advances in the study of wave propagation in random media have helped in the effort t o develop a unified scintillation 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 theory for strong scintillation has also made much progress in recent years but there still remains quite a few unresolved problems.

In this 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 scintillation will be the subject of another review paper [ 251 and will not be discussed here.

B. Motivation for Stochastic Formulation of the Problem Wave propagation is concerned with the study of the space- time fields that are transferred from one part of the medium t o another with an identifiable velocity of propagation. To identify the velocity of propagation, one may choose t o 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 its magnitude, change its shape, and even change its velocity provided this 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 00 18-92 19/82/0400-0324$00.75 0 1982 IEEE




0 10

5 eo

I S 3b

t S 35

4 0

io T

-10 4

0 4

40 lo

5 I S t0

es 30 1 5


I 3 .



2 .

- .

0 .

0 10 tO

5 I S L S 30

4 0 3s



-10 i 0

4 4 0 10

15 20

t S 30


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.



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 incident field uo satis- fies the equation Luo = 0.

iii)Shape and position of Boundary conditions 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 t o the usual deterministic approach in solving (1.1). Generally, a 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, instructive, 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 stochastic approach is then a useful approach.

In the stochastic approach one may classify the problem according to which one of the four quantities is stochastic.

Therefore, in studies of excitation of fields by random sources, the real source q is random; in studies of diffraction by partially coherent fields, the incident field uo 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 t o one of these four classes for their mixtures. According to this scheme of classification the study of ionospheric scintillations would normally belong to the class of problems dealing with diffraction and propagation through a random medium. However, under certain condi- 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: one proability space for the specification of the problem and one proability space for the wave field. A point in the probability space corresponds t o a particular probability distribution that is used t o charac- terize the problem or the field. Our interest 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 in the 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 obtained by 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 is interested in the behavior of radio rays, propagating in a fluctuating dielectric medium with certain statistical properties, one can first use the specified statistical properties t o realize many



Probabllity Space

of the Problem Probabllity Space

of the Wave Function

Fig. 2. A point in the probability space of the problem specifies the probability distribution of the dielectric permittivity or electron specifies the probability distribution of the wave function. Our density and a point in the probability space of the wave function interest is t o find the mapping between these two probability spaces as depicted symbolically by this illustration.


Fig. 3. A realization of a two-dimensional random surface with the prescribed statistical properties. (After Youakim e t a l . [27].)

6 I

4 1

/ I

- 4

-6 I I I I I I I I I I I I 1 1

5 10 I5

Fig. 4. Ray trajectories through realized dielectric media. All media a value of 1.5 percent in r m fluctuations of refractive index. Statisti- have identical power spectrum for the fluctuating refractive index and

achieving the mappingdepictedin Fig. cal properties of the rays can be compiled from these traced rays, thus 3. (After Youakim et al. [ 281 .)

media and then trace rays, all with identical initial conditions in these realized media. The results for one such study are shown in Fig. 4 [28]. The statistical behavior of the ray can be obtained if a sufficiently large number of such rays have been traced, as done in [28] 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



t o apply, and one would rather use an analytical method if it is available. At the present time, 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 such cases, approximations are often needed and introduced to facilitate a solution. The problem of ionospheric scintillations is no exception.


A . Observational Evidence

The existence of ionospheric irregularities is required to explain many experimental observations. The earliest is the vertical sounding experiment [30] in which a radar echo is received as the carrier frequency is swept from about 0.5 MHz to 15 MHz. The received data are typically displayed in the time delay (or virtual height) versus frequency format. Nor- mally the echo traces in such a display are very clean, showing distinct ionospheric layers. On occasion, the echo traces 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 t o study these spread F irregularities. A historical account of the experimen- tal effort can be found in [ 3 1

1 .

The experimental techniques can be broadly grouped into two: remote sensing techniques and in-situ measurements. Most remote sensing techniques utilize radio waves and they can be classified according to whether the radio waves are reflected from, scattered from, or penetrating through the ionosphere. In a low-power opera- tion the radio waves are normally reflected from the ionpsphere in experiments such as vertical ionosonde, backscatter ion@

sonde, and forward scatter ionosonde. Such experiments are useful in detecting the existence of spread F irregularities and their results have been used in morphological studies as re- viewed by Herman [ 3 2 ] . As the radio frequency is increased beyond some value, the radio wave begins 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 scattered back. Under quiescent condi- tions the backscattering is caused by ionospheric plasma fluc- tuations under thermal agitations. For sufficiently powerful

radars the scattered signal may be strong enough to provide us with useful information. Radars operating on this principle are known as incoherent scatter radars [33]-[35]. In a monostatic mode the backscattered power is proportional to the spectral content of electron density fluctuations at one-half of the radio wavelength. It must be understood, therefore, that such radars can sense the irregularities only in a very nar- row spectral window. On occasion, during the presence of spread F irregularities, the radar returns have been observed to increase in power by 80 dB in a matter of few minutes [ 3 6 ] . This means that in a few minutes the irregularity spectral

intensity can increase 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- tion measurements and is the subject of this review. Early reviews on this subject have been made by Booker [ 381 using



SCALE (km) MAGNETIC FIELD (m) 00 100 IO 1 100 IO I 0.1 0.01

1 I I I I I I 1 I

to Ionosphere Wanderlng of Normal

Multiple Normals

.- v


of TlDs Phose


(Gravitationally Sclntillatlon $

Anisotropic 1


t I

Strong Bac

Scottiring and Trans- equatorlal


WAVE NUMBER ( n i l )

Fig. 5. A composite spectrum summarizing intensity of ionospheric irregularities as a function o f wavenumber over a spatial scale from the electron gyro-radius t o the radius of earth. (After Booker [ 461 .)

radio stars as sources and by Yeh and Swenson [39] using radio satellites as sources. Because of the simplicity of experi- ments, the scintillation observations can be carried out at many stations. Globally it has been found that scintillations are most intense in two auroral zones and the magnetic equator

[ 161


Both the spectra of scintillating phase [40] and scintil- lating amplitude [41] have an asymptotic power-law depen- dence, This suggests that the ionospheric irregularity must have a power-law spectrum as well [42]


More recent progress on scintillation theories and experimental observations are

reviewed in later sections.

The other experimental technique has to do with measuring ionospheric parameters in situ. This generally implies carrying out measurements on board a rocket or a s'atellite. Probes have been made t o 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 the expectations based on the 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 [46] in a composite spectrum reproduced in Fig. 5. This composite




0045 0030







Fig. 6. Sample data of 136-MHz signals transmitted by the geostation- ary satellite SMSl parked at 90'W and received at Natal, Brazil (35.23OW, 5.8S'S, dip -9.6') on November 15-16, 1978. The bot- tom amplitude channel is approximately linear in decibels with a full scale corresponding t o 18 dB. The top and middle polarimeter out- full-scale change corresponds t o a rotation of puts vary linearly with the rotation of the plane of polarization. A 180' or a change of

1.89 X 10" el/m2 in electron content. The times given are in local mean time with UT = LMT + 03 : 00. Two successive depletions in electron content with accompanied rapid scintillations are sepa- rated by about 30 min in time.

spectrum spans an eight-decade range, corresponding t o scales from the electron gyroradius t o the earth radius. In this 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 equatorial scintillations associated with ionospheric bubbles. One example is depicted in Fig. 6, where the top 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 the bottom channel. While such bubble-associated scintillations are of great interest, we must remember 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 subject should consult a recent review [48].

B. Correlation Functions 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 t o describe the propagation problem stochastically as discussed in Section I-B. For this purpose we must first deyribe the medium, by its statistical properties. Thus let A N ( r ) be the fluctuations of electron number density from the background N o . Depend- ing on the problem, we may let


= A N ( ; ) or let = AN(;)/

N o ( z ) ; in either case is assumed t o be a homogeneous ran- dom field with a zero mean and a standard deviation ut. Its autocorrelation function is, by definition,

B E (;I


;2) = (E(;1) E(;2


(2.1) where the angular brackets are used t o denote the process of

ensemble averaging. By the Wiener-Khinchin theorem, the correlation and the spectrum form a Fourier transform pair





is real, there must exist symmetry conditions

BE (-;) = B E (;) and Qpg.


= ' D E (I?). (2.3) If the irregularities are $otrzpic, the correlation function in (2.1) depends only on ( r ,




In this case, the three-dimen- sional Fourier transform given in (2.2) simplifies to




@,(K)K sinKrdK. (2.4b)

r o

In some applications, the one-dimensional and two-dimen- sional spectra are needed and they are defined, respectively, by




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 or a satellite. However, the isotropic property is paramount in deriving the relation (2.7). In general when irregularities are anisotropic, it is impossible to deduce @ E


from V t ( K ~ ) .

In the ionosphere, probe measurements on board several earlier satellites have all yielded a power-law one-dimensional spectrum of the form V t a K;"' 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 70 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 [411, I 5 1 1 by using,the phase screen scintillation theory [42] or the Roytov solution [52].

There are indications, however, from recent multitechnique measurements, that the spectral index p may vary as the strength of the irregularities changes [ 1621. The power spec- trum maintains its power-law form to K


2 m-' (or spatial scale = 3 m) when the in-situ data are supplemented by the radar data at 50 MHz [ 531, [54]. There is indication, at least sometimes, that such a spectrum can be extended to irregular- ities as small as 11 cm 1551, [561. Nevertheless, on mathe- 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, the moments of (2.8) may not all exist; some of the integrals will diverge unless proper cutoffs are introduced. Physically, a departure from (2.8) is expected near an inner scale where dissipation becomes important and also near an outer scale at which the energy feeding the instability occurs. Recent rocket-borne beacon

experiments [ 211 and in-situ measurements [ 21 I ] covering more than five decades of scale sizes have shown a possible two-component power-law spectrum for the equatorial irregu- larities with a higher spectral index for the small structures.

To characterize the general power-law irregularity spectrum with spectral index p , Shkarofsky [57] introduced a fairly general correlation-spectrum pair

where ro is the inner scale and I o 2 7 7 / ~ ~ is the outer scale, and as such we must have KOrO


1 which is always implied.

Accordingly there exists a range of K values for which K O




l/ro and in this range the spectrum (2.10) simplifies to

which has the desired power-law form given by (2.8). For

K T o


1 , (2.10) reduces to

(2.12) which decays exponentially for large K . The correlation func- tion (2.9) has the desired properties in that, at the origin = 0, B E has a maximum value, a vanishing first derivative, and a negative second derivative as discussed by Shkarofsky [ 571.

The corresponding one-dimensional spectrum can be obtained by substituting (2.9) into (2.5). The integral can be evaluated exactly to give

It can be shown that the three-dimensional spectrum (2.10) and the one-dimensional spectrum (2.13) satisfy the relation (2.7). For K O




l/ro, V t reduces to

a- 1

K X P - 2 (2.14)

which follows also a power law but with a spectral index p - 2 instead of p as is the case for @'E.

Generalization to the anisotropic case can proceed in the fol- lowing way. Introduce the dimensionless scaling factors a,, ay, and a, along the three axes. The correlation function (2.9) is modified by replacing rz by x z / g + y 2 / a ; +z2/a:.

Accordingly, modifications on the three-dimensional spectrum involve the multiplication of (2.10) by a a a and the re- placement of K' by a ; ~ : +CY;K;

+ g ~ : . ,

Smdarly, . y . 2 the one-

dimensional spectrum can be modified by multiplying (2.13) by a, and replacing K: by


K:. These modifications can be easily introduced [71, [801, [991.

C. Optical Path and Correlation of the Total Electron Content

In the homogeneous background for a ray initially pointed along the z-axis, the fluctuations in the optical path are given by

W P ' ) =


2) dz (2.15)



= ( x , y ) is the transverse coordinate and the integral is carried out from some initial point to the final point. In the ionosphere, the refractive index fluctuations A n are caused by the electron density fluctuations AN through an approxi- mate linear relation under the high-frequency approximation.

Accordingly, the deviation of the optical path from the mean can be expressed as



(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 quantity A N , is the devia- tion in the total electron content defined by

AN,( p') = A N ( p', z) d z . (2.17) The correlation of the optical path separated by a distance




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)


where K~ -b = ( 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 as shown. Inserting (2.19) into (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.,




BANG, z) dz. (2.20)

Consequently, the electron content correlation is merely the product of the propagation path z and the integrated correla- tion function (2.20). For the three-dimensional correlation function given by (2.9), A is found t o be


(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 - ~as 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 fluctuations 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 more convenient t o deal with 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 t o be

(2.24) for path lengths z greater than the correlation length as is usually the case. The optical path structure function is there- fore directly proportional t o the electron content structye function. If ANis apmogeneous random field, thenDAN ( r ) = 2 [BAN(O)


BAN(^)] which reduces (2.24) t o

where the optical patn structure function is simply related t o 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


= [ ( 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, as postulated 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 the Doppler broadening. For the moment if we take only the convection into account, the random field then satisfies

E(;, t


t ' ) = ((;- ZOt', t) (2.28) for which the space-time correlation has the form

B E ( ; , t) = E t ( ; - &t). (2.29) In (2.28) and (2.29),


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 fields, the correlation function satisfies (2.29) and their space-time spectrum satisfies

If this frozen field is also isotropic, we can show that



where W E (a) is the frequency spectrum on a time series .$(;, t ) obtained by a fixed observer. The prime on W indicates dif- ferentiation. Equation (2.31) relates the spatial spectrum t o the frequency spectrum of an isotropic frozen random field.

When the spectrum is generalized to include nonfrozen flows, we must take into account the possibility that irregular- ities may change with time as they move. In doing so it is

desirable to strike a balance between a reasonably simple

analytic expression that can be manipulated mathematically and the physical notion that 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 in the following way:

S E G , a) = $ G w ) (2.32) with the normalization


$(;, w ) dw = 1. (2.33)

In the interest of not flooding this review paper with too many symbols, let the+argument of $ denote the Fourier domain.

For example $( K , t ) is obtained from $ (I?, w ) by 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 $(


w ) = 6 (w

+ I? -



+ +

$(;, 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


If the decay is caused entirely by velocity fluctuations with a standard deviation uu, (2.35) can be generalized to

(2.36) The frozen field result of (2.35) is obtained from (2.36) for large irregularities (viz., small K ) and short time as is desired based on physical reasoning discussed earlier. By Fourier transforming (2.36) with respect to t and substituting the result in (2.32), the space-time spectrum becomes

and the corresponding correlation function becomes


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


in the exponent in (2.38) makes contribution to the integral 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 BE ,(; t ) must have the asymptotic behavior t-’ for large times, The velocity


in (2.38) does not necessarily have to be the convective velocity of the fluid. In measurements made in situ

by probe carrying satellites and rockets,


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

is,then BE ( r ( t ) , t ) where r ( t ) = G o t 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 is termed the ambiguities of deducing the rest frame irregularity spectrum from the moving frame spectrum. Let PE (a) be the spectrum deduced in the moving frame, viz.,


PE (a) = (27r)-1Im BE (;(r), t ) ,-jut d t . (2.39)

For a rectilinear motion of the probe we may :et ; ( t ) =z^uot where


is a unit vector along the z-axis. Since a satellite travels with large velocities, the random field as observed by+the probe may be approximated as frozen. Consequently, BE (r ( t ) , t ) = Bg(z^uot) which when inserted into (2.39) yields


where V E is the one-dimensional spectrum defined in (2.5).

Therefore, the moving frame spectrum P E ( ( ~ ) is related to the one-dimensional rest frame spectrum V E ( C , 0, K , ) by (2.40) with K , = a / u o under the frozen field assumption. If the frozen field is isotropic, the deduced one-dimensional spec- trum can in turn determine the three-dimensional spectrum by using (2.7). If the frozen field is anisotropic and of the kind discussed at the very end of Section 11-B, the three- dimensional spectrum can be recovered only when we also know g,, a,,, and the orientation of the probe motion relative to the correlation 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

BE = e-‘ 2I r 2 (2.42)

then the integral in (2.41) can be integrated t o give

(2.43) Hence when t <<I&, (2.43) reduces to which is the one-dimensional correlation function along the path of a moving probe, in agreement with (2.40). Notice that the frozen field is valid only for times short compared with the time required to move through the irregularity with an rms velocity. In the other extreme when 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) is difficult to evaluate analytically. The moving frame spectrum Pc(w) in this general case is related

2 2 2





lO.O.2 I



Fig. 7. Geometry of the ionospheric scintillation problem.

to the rest frame spectrum @E




(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 t o invert (2.44) to get @E


with- out making additional assumptions.

111. SCINTILLATION THEORIES A . Statement of the Problem

With the statistical characterization of the irregularities as discussed in Section 11, we can model the ionospheric scintilla- tion phenomenon. Let us consider the situation shown jn Fig. 7.

A region of random irregular electron density structures is located from z = 0 t 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




(3.1) where ( E ) is the background average dielectric permittivity which for the ionosphere 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 the back- ground electron density N o and f is the frequency of the inci- dent wave. In the percentage fluctuation A N / N o =


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 flow, or both, are assumed t o be much slower than the period of the incident wave.

As the wave propagates through the irregularity slab, t o 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@ is the optical path fluctuation 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 p r o p agates to the ground, the distorted wave front will set up an interference pattern resulting in amplitude 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 =


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 t o focus or de- focus the rays. However, when the irregularities are strong such that el 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 t o focus or defocus the rays. The interference scenario for the amplitude fluctuation described above therefore is no longer valid. Qualitatively, one would expect the 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 amplitude 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 to investigate quantita- tively the various aspects of the phenomenon. The starting point is the wave equation in electrodynamics. Under the assumptions [ 5 8 ]

i) the temporal variations of the irregularities are much ii) the characteristic size of the irregularities is much greater the vector wave equation for the electric 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. Unfortunately, the general solution of (3.4) does not seem t o be possible. One has t o settle for various approximate solutions for different applica- tions. T o discuss these solutions, we f i t specialize 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 the wave 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 t o treat prob- lems of wave propagation in random media [ 241. The follow- ing assumptions are made in this approach:

iii) The Fresnel approximation in computing the p h g e of the scattered field is valid, corresponding t o z



>> h

iv) Forward scattering: The wave is scattered mainly into a small angular cone centered around the direction of propagation. This corresponds to ( e t ) z / l < < 1, where z



is the distance the wave has traveled in the random me- dium and 1 is the characteristic scale of the irregularities, which can be taken as certain mean scale size of the irregularities [ 241, [213]. In addition, the backscat- tered power is negligible, corresponding t o ( E : ) kz



v) The attenuation of the coherent wave field per unit wavelength is small, corresponding t o ( E : ) k2



When assumption iii)is satisfied, it follows that (3.5) can be approximated by

- 2 j k - + + f ~ = - k ~ ~ l ( ; ) ~ ,

a U

O < Z < L (3.6)


where 0: =


/a x z

+ a2

l a y2 is the transverse Laplacian. This is an equation of the parabolic type whose solution is deter- mined uniquely by the “initial” condition at z = 0. This equa- tion has been used in quasi-optics and other propagation prob- lems [641. Based on (3.6), with the additional assumptions iv) and v), a series of equations for the moments of the com- plex amplitude can be derived that constitute the basis for the scintillation theory. Below the irregularity slab, under the assumption that the scales of the random variation of the field are large in comparison with the wavelength, the complex amplitude satisfies

- 2 j k - + V f u = O ,

a U

z > L .



The “initial” condition for (3.7) is given by the solution of (3.6) evaluated at z = L .

Therefore, (3.6) and (3.7) are the basic equations upon which the ionospheric scintillation theories are developed. In the following, we shall discuss several such theories.

B . Phase Screen Theory

Historically, the first ionospheric scintillation theory was based on the idea of wave diffraction from a phase-changing irregular screen [221, 1231, [65,1-[731. This idea has been qualitatively discussed in the previous section. Let us consider an incident plane wave with constant amplitude A o . As the wave passes through the irregularity slab, the ionosphere acts as a phase-changing screen that modifies only the phase of the wave. Therefore, upon emerging from the ionosphere, the wave has the form

uo(P’> = ~0 exp [ - M A I . (3.8) The irregularity slab is considered to act as a thin screen located at z = 0 that contributes to changing the phase of the incident wave by the amount

where re = eZpo 4nm is the classical electron radius and A N T ( p’) is the deviation of the total electron content through the irregularity slab.

As the wave u 0 ( d ) propagates to the ground, the field can be computed using the Kirchhoff’s diffraction formula [ 231.

Under the forward scattering assumption, the Fresnel diffrac- tion results in [ 231

(3.10) It is interesting t o note that (3.10) satisfies (3.7) which is the equation governing the wave propagating below the irregularity slab under the forward scattering condition.

Equation (3.10) is the starting point of the phase screen theory for ionospheric scintillation. To develop the theory, one assumes that the phase perturbation introduced by the screen is a Gaussian random field with zero mean. It is rea- soned that the contribution to the phase fluctuation comes from many irregularities along the line of sight. The central limit theorem then predicts the Gaussian distribution for the phase. Utilizing the property

for the homogeneous Gaussian field @, we obtain from (3.10) the expression for the mean field on the ground

( u ( $ , z ) ) = A ~ exp [-&/21 (3.12) where

- * +”


(3.13) We have used (2.19) in deriving (3.13) and L is the thickness of the irregularity slab; @ A N ( ; L , 0) is the three-dimensional spectrum of the density fluctuation AN with K , set equal to zero.

The averaged field is attenuated according t o (3.12). This is due t o the fact that part of the energy has gone t o the inco- herent part of the total field which is generated by the random phase front.

The average intensity on the ground can also be computed from (3.10)

(u(p’,z)u*(;,z))=A; (3.14)

which is a constant equal to the incident intensity, consistent with the forward scattering assumption.

In the experimental observation of the scintillation phenom- enon, one often measures the fluctuations of the amplitude (intensity) of the received signal. In recent years, thanks to several satellite beacon experiments 1201, [ 741, the phase fluctuations of the signal can also be simultaneously measured.

Therefore, it is of interest to derive useful formulas for these observed quantities. In order t o facilitate comparison with results from other scintillation theyies, we proceed in the following manner. Let the field u ( p , z) in (3.10) be written in the form

u(p’, z) = A0 exp [x(?, 2) -

is1 (3,

z)l = A0 exp W G , z)l (3.15) where x(;, z) is referred t o as the log-amplitude and S1


z )

as the phase departue of the wave.

For a “shallow screen” such that @:


1, it is easy t o &ow from (3.10) that


(3.16) From (3.16) we obtain the following results for the moments





and S1 : The mean


= (SI) = 0. (3.17) The correlation functions

(3.18) where @@(


is the power spectrum for the phase q5( p’) given by

@ ~ ( $ ~ ) = h 2 r ~ @ A N T ( $ ~ ) = 2 ~ L h 2 r ~ @ A N ( ~ ~ , 0).

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


and S1

/-r +-

(3.20) and the power spectra for the log-amplitude and the phase departure


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

= 2nLh2r,? S h 2 ( K f Z / 2 k ) @ ~ ~ ( 2 1 , 0)


= COS2 (K:Z/2k)@,#,($l)

= 2nLh’r: COS’ ( K : Z / ~ ~ ) @ A N ( $ ~ , 0). (3.21) As mentioned above, the phase screen theory has been used quite extensively in ionospheric scintillation work as well as interplanetary and interstellar scintillations [4], [75]-[77].

Although the derivation was specialized for an incident plane wave, the results can be readily generalized t o 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 t o go back t o (3.10) t o derive general expressions for the various parameters. Mercier [69] considered this problem in some detail and derived integral expressions for the higher moments of the field. Recently, several authors have derived analytic asymptotic expressions for the intensity correlation function and the spectrum [ 801 -[ 861. Some of these results will be discussed in later sections.

C. Theory for Weak Scintillation-Rytov Solution

When the effects of scattering on the amplitude of the wave inside the irregularity slab are t o be included in the treatment of the scintillation phenomenon, one has t o go back to (3.6) and (3.7). With the substitution of (3.15), (3.6) becomes

Under the assumption of weak scintillation such that the higher order term


can be neglected in (3.22), we ob- tain the equation for the Rytov solution [ 241


[ 581

(3.23) The range of validity of this solution has been discussed by many authors [87]-[89]. There is some evidence that the Rytov solution may be applied t o ionospheric scintillation data even for moderately strong scintillations [go].

The general solution of (3.23) can be obtained as

exp [-jkl;




f)1 d’p’ (3.24) where $ o ( p ’ ) =


u(p’, 0) corresponds to the incident wave.

The field emerging from the bottom of the slab is given by exp [


(p’; L)] , which contains modifications for both ampli- -tude and phase. The amplitude variations come about from the diffractional effects inside the slab, as is evident from the second term in (3.24). The field on the ground 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)1 d’p’ (3.25) where $(p’f, L ) is obtained from (3.24).

Equation (3.25) gives the formal solution for the ionospheric scintillation problem under the Rytov approximation. It can be used t o derive the various statistical parameters for the wave field.

Again, let us specialize t o a plane incident wave with unity amplitude. Then the mean values


= (Sl> = 0. The power spectra for


S 1 , and the cross spectrum between


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

?rk3 KfL K ?

@,s(K;) = 7 sin






L/2)@&, 0).

K l 2k k

(3.26) The correlation functions can be obtained from (3.26). We note that by letting L + 0 in the expressions for @,,


we obtain the phase screen results (3.21) if the substitution = (r:h4/nz) @ A N is made.

Several aspects of this result are specially useful in the anal-




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