Radio Wave Scintillations in the Ionosphereb92b02053/printing/summer/Materials... · YEH AND LIU:...

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

Radio Wave Scintillations in the Ionosphere

KUNG CHIE YEH, FELLOW, IEEE, AND CHAO-HAN LIU, FELLOW, IEEE

Invited Paper

Absfruct-The phenomenon of scintillation of radio waves propagating 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 in-siru, 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 scintillation 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, rms phase fluctuations, correlation functions, power spectra, distriiutions, etc., are investipted. Effects of multiple scattering are discussed. 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 temporal 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.

I. INTRODUCTION

I 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 I -[ 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 to 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 atmosphere. 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 observations. 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 to 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 scintillation of radio waves will be reviewed, both from the observational 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 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 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 frequently an integro-differential operator or a tensor operator

00 18-92 19/82/0400-0324$00.75 0 1982 IEEE

YEH AND LIU: RADIO WAVE SCINTILLATIONS IN THE IONOSPHERE 325

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eo IS

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4 0

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5 I S t0

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i

2 .

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0. 0 10 tO

5 I S 30

LS 40

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, i

-10 i 0

4 4 0 10

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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.

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 to specify:

i) Real source function q : Usually localized in space and time.

ii) Virhial source function u o : The incident field uo satisfies 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 quantities may become very complex. When this is so the wave function is also expected to 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, 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 obtained 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 characterization yields a stable and physically meaningful statistical 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 to 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 conditions and sometimes in an effort to simplify the mathematical task, the phase screen idea is advanced. In this case the problem 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 to a particular probability distribution that is used to characterize the problem or the field. Our interest in solving (1.1) is then to find the prescription that maps a point in the probability 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 probability .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-dimensional random surfaces can be generated [27] , 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

1 Mopplng

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 to find the mapping between these two probability spaces as depicted symbolically by this illustration.

H CORRELATION LENGTH

Fig. 3. A realization of a two-dimensional random surface with the 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 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. 3. (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 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


to 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 distributions, 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.

11. CHARACTERIZATION OF IONOSPHERIC IRREGULARITIES

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 ionospheric 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 experimental 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 reviewed by Herman [32] . 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 conditions the backscattering is caused by ionospheric plasma fluctuations 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 [36] . 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 certain type of spread F irregularities take the form of plumelike structures and may be caused by Raleigh-Taylor instabilities [37] . Another remote sensing technique deals with scintillation measurements and is the subject of this review. Early reviews on this subject have been made by Booker [ 381 using

HORIZONTAL SCALE I 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 e

of TlDs Phose e (Gravitationally Sclntillatlon $ Anisotropic 1

H

t I Strong Bac Scottiring and Trans- equatorlal

(Magneticoily

WAVE NUMBER (ni l ) Fig. 5. A composite spectrum summarizing intensity of ionospheric

irregularities as a function of 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 experiments, 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 scintillating amplitude [41] have an asymptotic power-law dependence, 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 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 [43] . The power spectrum of A N is found to follow a power law [44] , [45] , 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

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

- 0045 0030

I

0015 I

0000 I

2345

Fig. 6. Sample data of 136-MHz signals transmitted by the geostationary 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 bottom 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 local mean time with UT = LMT + 03 : 00. Two successive depletions in electron content with accompanied rapid scintillations are separated by about 30 min in time.

spectrum spans an eight-decade range, corresponding to scales from the electron gyroradius to the earth radius. In this seven- decade range, irregularities responsible for i:nospheric scintillations vary from meters to 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 channel, and the depletion in TEC by 5.7 X 10l6 el/m2 as indicated 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 ionization 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 g = AN(; ) or let = AN(;)/ N o ( z ) ; in either case is assumed to be a homogeneous random field with a zero mean and a standard deviation ut. Its autocorrelation function is, by definition,

BE (;I - ;2) = (E(;1) E(;2 1) (2.1)

where the angular brackets are used to denote the process of

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

m

(2.2b)

Since .$ is real, there must exist symmetry conditions

BE (-;) = BE (;) 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-dimensional Fourier transform given in (2.2) simplifies to

m

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

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

OD


For the special case of isotropic irregularities, the three-dimensional 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 measurement 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 (2) 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 Vt a K;"' with m close to 2 [44] , [49] , 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 [40] , [SO] and of amplitude [411, I511 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 spectrum 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 irregularities as small as 11 cm 1551, [561. Nevertheless, on mathematical 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 irregularities 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 << K << l/ro and in this range the spectrum (2.10) simplifies to

which has the desired power-law form given by (2.8). For KTo >> 1 , (2.10) reduces to

(2.12)

which decays exponentially for large K . The correlation function (2.9) has the desired properties in that, at the origin = 0, BE 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 << K << l/ro, Vt reduces to

1 a-

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 following 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 replacement 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 4 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 J- (2.15)

where p' = ( 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 approximate 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 deviation in the total electron content defined by

AN,( p') = AN( p', z) dz . (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 ~ , 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 directly 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) dz. (2.20)

Consequently, the electron content correlation is merely the product of the propagation path z and the integrated correlation function (2.20). For the three-dimensional correlation 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 - ~ 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 to be

(2.24)

for path lengths z greater than the correlation length as is usually the case. The optical path structure function is therefore directly proportional t o the electron content structye function. If ANis apmogeneous random field, thenDAN ( 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, as postulated in [59] and [ 601, are caused by 1) the convection of scattering irregularities 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), 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 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 to 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 irregularities 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 decompose the spectrum S in the following way:

SEG, a) = $ G w ) (2.32)

with the normalization

I: $(;, 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-dimensional Fourier inversion with respect to w. With such a notation, the spectral decomposition scheme (2.32) plus the normalization (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 + I? - 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 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

00

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 Go 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, So becomes the velocity of the p:obe. The co2elation function of such in-situ data i s , then BE ( r ( t ) , 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 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.,

00

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 z ̂ 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

(2.40)

where VE 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 VE(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 spectrum 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

B E = e-‘ I r 2 2

(2.42)

then the integral in (2.41) can be integrated to 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 correlation (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

3 3 2 PROCEEDINGS OF THE IEEE, VOL. 70, NO. 4, APRIL 1982

TO TRANSMITTER AT-CD

I

lO.O.2 I - RECEIVER

Fig. 7. Geometry of the ionospheric scintillation problem.

to the rest 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 @E (2) without 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 scintillation phenomenon. Let us consider the situation shown jn Fig. 7. A region of random irregular electron density structures is located from z = 0 to 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 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 background electron density No and f is the frequency of the incident wave. In the percentage fluctuation AN/No = 5, the temporal variations, caused by either the motion of irregularities as in a frozen flow or the turbulence evolution as in a nonfrozen flow, or both, are assumed to be much slower than the period of the incident wave.

As the wave propagates through the irregularity slab, to the first order, only the phase is affected by the random fluctuations 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 = d-, which is the size of the first Fresnel zone [63] . 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 defocus the rays. However, when the irregularities are strong such that el is relatively large, the phase deviations may become so intense that the phase front is no longer coherent across the irregularities larger than certain size. These irregularities then lose their ability to 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 irregularity slab is thick one would expect to see amplitude fluctuations 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 k2 = 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 to be possible. One has to settle for various approximate solutions for different applications. To 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 introduce the complex amplitude for the wave field u ( r )

Equation (3.4) then yields an equation for the complex amplitude

Based on this equation, an approach, known as the Parabolic Equation Method (PEM), has been developed to treat problems of wave propagation in random media [ 241. The following assumptions are made in this approach:

iii) The Fresnel approximation in computing the phge 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 the direction of propagation. This corresponds to ( e t ) z / l < < 1, where z


is the distance the wave has traveled in the random medium 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 backscattered power is negligible, corresponding to ( E : ) kz << 1.

v) The attenuation of the coherent wave field per unit wavelength is small, corresponding to ( E : ) k2 << 1.

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

- 2 j k - + + f ~ = - k ~ ~ l ( ; ) ~ , O < Z < L (3.6) a U a Z

where 0: = a2 /axz + a2 lay2 is the transverse Laplacian. This is an equation of the parabolic type whose solution is determined uniquely by the “initial” condition at z = 0. This equation has been used in quasi-optics and other propagation problems [641. Based on (3.6), with the additional assumptions iv) and v), a series of equations for the moments of the complex 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 jk -+Vfu=O, z>L . a U aZ (3.7)

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 ANT( 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 diffraction results in [ 231

(3.10)

It is interesting to 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

- * +”

-00

(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 to (3.12). This is due to the fact that part of the energy has gone to the incoherent 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 phenomenon, 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 to as the log-amplitude and S1 (3 , z ) as the phase departue of the wave.

For a “shallow screen” such that @: << 1, it is easy to &ow from (3.10) that

A”

(3.16)

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


of x and S1 : The mean

(x> = (SI) = 0. (3.17)

The correlation functions

(3.18)

where @@( zl) 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 fluctuations for x and S1

/-r +-

(3.20)

and the power spectra for the log-amplitude and the phase departure

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

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

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

= 2nLh’r: COS’ (K:Z/~~)@AN($~, 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 to cases of spherical 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 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 to be included in the treatment of the scintillation phenomenon, one has to 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 (VI$)’ can be neglected in (3.22), we obtain 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 to ionospheric scintillation data even for moderately strong scintillations [go].

The general solution of (3.23) can be obtained as

exp [-jkl; - p”I2/2(z - f)1 d’p’ (3.24)

where $ o ( p ’ ) = In 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 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 condition. The Rytov solution for (3.7) is

exp [-jk l p ’ - p” 1’/2(z - 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 to derive the various statistical parameters for the 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, S1, and the cross spectrum between x and SI for the field on the ground are given, respectively, by [ 9 1 ]

?rk3 KfL K ? @,s(K;) = 7 sin - sin - (z - 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 @,, @s, we obtain the phase screen results (3.21) if the substitution = (r:h4/nz) @AN is made.

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


0 2 4 6 0 10 I? 14 16 18 20

Fig. 8. Filter function for amplitude scintillation plotted 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 scintillation index defined as the normalized variance of intensity of the signal [ 9 1 ]

( I Z ) - sf = . (3 .27)

Other definitions for the scintillation index have been proposed [ 9 2 ] , [ 9 3 ] , [44]. However, the S4 index has been adopted by most investigators for digitally processed scintillation data. For weak scintillations, it is easy to show [ 9 1 ]

sf = 4(X*). From (3 .26) and the definition of correlation function, we have

This quantity measures the severity of intensity scintillation under the weak scintillation assumption. The integral in (3 .28) indicates that the contribution to the intensity scintillation from the irregularities is weighted by a spatial filter function, i.e., the expression in the square brackets of (3 .28) . Fig. 8 shows the filter function versus K ' ( Z - L ) / k for three values of the slab thickness L . The height of the slab is 350 km. The oscillatory character of the filter function is known as the Fresnel oscillation, which is more pronounced for a smaller L . The irregularity spectrum is, in general, of a power-law type, which decays as K increases. Therefore, the product of the filter function and the spectrum has a maximum around K fi K F = 2 n / d ~ , corresponding to the first maximum of the filter function. This is consistent with the intuitive picture presented in Section 111-A that when multiple scattering effects are not important, irregularities of sizes of the order of the first Fresnel zone are most effective in causing amplitude scintillation.

For a power-law spectrum QAN - K I P , with an outer scale much greater than the Fresnel zone size, it is possible to show from (3 .28) that [ 5 2 ]

s4 a ~ ( 2 + ~ ) 1 4 a f - ( 2 + ~ ) / 4 . (3 .29)

This frequency dependence of the scintillation index has been observed in many experiments, some of which will be discussed in later sections.

From (3 .28) we also note the dependence of the scintillation index on the thickness of the slab L ' I 2 , and on rms AN.

2 ) Mean-Square Phase FZuctuations: From ( 3 . 2 8 ) we have

..

The phase filteMg function given in the square brackets of (3 .30) is very different from the amplitude filtering function, which as discussed in the last section shows Fresnel effects. In fact, the major contribution to (St) comes from the large irregularities. It is easy to show from (3 .30) that (St) is proportional to l / fz .

3) Frequency Power Spectra: In practical situations, the irregularities in the ionosphere are in motion most of 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 radio signals transmitted from the geostationary satellite, this 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 the spatial behavior. In other cases where the radio signals are transmitted from a transit satellite, 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 variations of frozen irregularities. In both cases, the relation between temporal and spatial variations is a simple translation by the motion. The frequency power spectrum of the signal received at a single station denoted by @(a) is related to the spatial power spectrum by [ 5 1 ]

. ,-+-

where the coordinate system is chosen such that the drift velocity is in the x-z plane with the tranverse (x direction) speed uo .


Substituting 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 can be estimated. At the high- frequency end such that 52 >> 5 2 ~ = V ~ K F , both 9, and vary asymptotically as a('-P)

And for 52 << 52,

(3.34)

*OD

(3.35)

For the amplitude spectrum ax, the two asymptotes meet as

I is the ratio of the two integrals in (3.33) and (3.34). There- fore, on the log-log plot of the spectra, one observes the high- frequency asymptotes having a slope of (1 - p ) . At the low- frequency end such that 52152~ << 1, the amplitude spectrum approaches a constant independent of 52, while the phase spectrum s t i l l has an asymptotic slope of (1 - p ) . The rolloff for the amplitude spectrum occurs near 52 = 52, - 52~. This again reconfirms the intuitive picture that irregularities of sizes greater than the size of the f i t Fresnel zone do not contribute much to the amplitude scintillation. We note from this result that large-scale irregularities dominate the phase fluctuations.

For a Gaussian irregularity spectrum, the behavior of 9, and !& are quite different [95]. Indeed, the investigation of the shape of the observed scintillation spectra has played an important role in determining the power-law nature of the ionospheric irregularity spectrum [41], [ 421. The above are the basic results for weak scintillation under the assumptions of Rytov approximation. There is experimental evidence that the Rytov results for the amplitude scintillation remain valid for S4 approaching a value of 0.3, while phase fluctuations have a wider range of validity [ 901.

The above derivations are based on the normal incidence of the wave upon the irregularity slab. The results can be generalized to cases of oblique incidence [70], [96]-[991. It can be shown that under general ionospheric conditions the effects of diffraction are not altered by the orientation of the scattering

layer relative to the direction of propagation. The s c a n effect produced by the relative motion of the source and the irregularity slab acts to select only spatial frequencies normal to the line of sight [46]. That is to say that instead of the irregularity sizes in the x-y plane as appeared in the above derivations, it is now the dimensions of irregularities transverse to the line of sight that enter in the scintillation formulas. This results in the replacement of th+e irregularity spectrum @ A N ( K ~ , 0) by +AN (21, -tan e i?kl ~ 1 ) in the expressions derived, where e is the incidence angle of the wave (with respect to the vertical axis) and i?kL is the unit vector transverse to the propagation direction [991. In addition, the slanted path of propagation should be taken into account. The geometric dependence of scintillation becomes more important when the irregularities are anisotropic. With the combined effects of propagation geometry and anisotropy of the irregularities, enhancement of scintillation can sometimes result [ 1001.

The above derivations can also be extended to the case of spherical waves [ 10 1 1 -[ 103 1.

D. Parabolic Equation Method

The Rytov solution, in essence, is a single-scattering solution for the scintillation problem. As el (the fluctuation in the relative dielectric permittivity of the medium) increases, multiple scattering by these irregular structures becomes more important. For a given frequency, el increases with increasing density fluctuation AN. In a &en ionospheric condition, el increases for waves at lower frequencies. In either case, when multiple scattering effects become significant, scintillation theory based on Rytov solution will no longer be applicable. Indeed, the assumption of small phase fluctuations is often not satisfied if the data interval is large. One has to go back to (3.6) and (3.7) and develop a theory free of the Rytov assumption of smallness in fluctuations of the field quantities.

The development of such a scintillation theory is based on the realization that i) under the forward scattering assumption the field at any height z' depends only on those irregularities in the region z <z ' and ii) as the wave propagates in the random medium for a distance much greater than the longitudinal correlation distance of the e l , the field varies only a small amount in a correlation scale of el in the z direction. There- fore, for the computation of the statistical moments of the field, one can approximately replace the correlation function for the irregularities by

where

+m +m

A , ( ; ) =IOD B E ( ; , f) df = 2nJ 0) ei ' l . 'dZKL -m

(3.37)

is the two-dimensional correlation function. This is the so- called Markov approximation [ 581, [ 661.

Starting from the parabolic equations (3.6), with the Markov approximation (3.36), it is possible to derive a closed set of equations for the statistical moments of the field u , by using functional derivatives or diagram techniques [ 1041, [ 1 OS].

Let us now introduce the general moments for the complex amplitude u(p', z , k ) where the frequency of the wave is explicitly denote? by the paramete: k . For convenience, we introduce s' = ( p , k ) and express u(s , z ) as the field. The mnth


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

arm*n (z, a, . . . + . s“ . . +’ a 2

S m , ‘Sn)

(3.39)

v; = a2/ax; + a21ay,? and vj2 = a2/axjZ + a2/ayj2. For z > L , i.e., outside the 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 statistical moments of the scintillation signal. This set of equations was first used to develop a multiple-scatter scintillation theory for the ionmphere case 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 z > L

( U ) = A ~ exp [ - ~ ~ A ~ L A ~ ( o ) / ~ I = A ~ exp [-&/21.

(3.41)

This agrees with the plane wave solution from the 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 shall present some results obtained from the scintillation theory based on (3.39) and other equivalent versions of it. Emphasis will be on quantities that are observed in the scintillation experiments.

1) Mutual Coherence Function: Consider 1’1,1 = (u (z , p’, k)u * (z, z‘, k’)). The equation for rl, becomes

(3.42)

rl.,l is known as the two-frequency two-position mutual

&=1.6%9 C = I 5 5

c =2.97

-. - - GAUSSIAN POWER LAW

X

i i

01 0 2 03 5 4 0 5 0 6 0 7 08 09 I O

P / f O

Fig. 9. Contours of constant correlation coefficient C,,, for frequency- space separations. Both power-law and Gaussian irregularities are included.

coherence function [ 105 1. 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 ) ) which, 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 = (u2 ) = A: from (3.43) which is consistent with the energy conserva- tion requirements for forward scattering.

A Rytov type of solution for 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 approximation, we have [ 1081, [ 1091

rl, (z, 6, ?’, k, k’) = ~ X P [$I (3.44)

{exp [jAkKf(z - L)/2kk’] - exp [ j # r ~ f z / 2 k k ‘ ] }

’ 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 [ 1 101 and to characterize the transionospheric communication channel [ 1 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 integration. Fig. 9 shows some results from such computations [ 1 131. In transionospheric communication applications, it is useful to define a correlation coefficient for the complex amplitude

I((u - ( u ) ) (U’* - ( U ‘ * ) ) ) I

I ( Iu - ( u ) 1 2 ) (Iu’ - (u’)2)l 1’2

[( (u2> - (u>2) ((Ut? - (u’>2)] lI2 *

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


C = 8nl:rzX ( (AN)’) to = L/k , l$

( = z/kol $ A k X=- k ’ + k (3.47)

where lo is some characteristic size for the irregularities. The contours in Fig. 9 indicate the level of correlation for

si’g”+& at different frequencies received at stations separated by p. They can be used to study frequency and/or space diversity schemes for transionospheric communication. The results are given for both Gaussian and power-law irregularity spectra. Recently, using data from Wideband Satellite, the two-fre quency coherence function has been measured experimentally [2121.

2) Scintillation Index: r2,’ computed for the same frequency corresponds to the coherence function known in the literature =r4 =r2,2 =(u(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 index 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 ) is the structure function for the phase. ( = Z / L T 41 = /IT & = ;2 /IT q = KllT +

Introducing new variables -+ (3.55) R = + (Jl +& +& +&) ;1 = (;l - p’z +p’; - 3;)

and P’=j& + p ’ z - p’; -& i1 = 3 (51 1 - p”l

IT = (87~’rzC&X)-(’/~) (3.49) LT = [8n2(2n)-P/’ r ~ C & k ” ~ ~ Z ) ] - ( ’ ~ p ) . (3.56)

(3.48) can be transformed to For z > L , the dimensionless equation is

(3.50)

where F is the expression in the curly brackets in (3.48) involving 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 the 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. Equation (3.50) then becomes

In terms of the power spectrum, the F function can be expressed as

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

+-

-OD

* (1 - cos 21 ;2) d’K1 (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 ~ 1-P (3.53)

(3.5 7)

with r4 at z = L computed from (3.56) as its “initial” condition.

Since (3.54) and (3.57) are dimensionless, their solutions must be independent of the irregularity strength and the geometry. Indeed, it will be possible to obtain a universal solution for the problem [114], [1151. Equations (3.54) and (3.57) constitute the basis for multiplescatter ionospheric scintillation theory for intensity scintillations. Once r4 is known, one can compute the scintillation index S4.

s t = - r4((, 0, 01 - 1. (3.58)

This indicates that the scintillation index is a function of ( = z / L T . From (3.28), it can be shown that the scintillation index S40 computed under Rytov approximation (the sub- script 0 indicates the Rytov solution) is proportional t o (. This implies that the scintillation index in the general case is a function of S40 [ 1 151.

General analytic solutions to (3.54) and (3.57) have not been found although certain asymptotic solutions have been obtained [85], [ 1161. Numerical solutions have been attempted for some cases [ 1161, [ 1171. Fig. 10 shows the results from one of such computations [53]. The scintillation

1

A40


2’ow

0.0 0.0 0.1 0.2 0.3 a4 0.5

S, AT 5 0 0 MHz

Fig. 11. Spectral indices for two frequencies against scintillation index S, . The ionosphere conditions are the same as those in Fig. 10.

2 L=SOkrn fP=707MHz 3. L=50krn f,=IOMHz

Frequency f (MHz)

Fig. 12. Spectral index as a function of frequency for different iono- 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 frequencies increases linearly with ANrms, as predicted by the weak scintillation theory (3.28). As ANrms increases, md- tiple scattering effects become important, and saturation of the scintillation index becomes apparent, first at the lower frequency. This saturation effect causes the frequency dependence of S4 t o depart from that predicted by the weak scintillation theory, viz., S4 af-”, n = -(2 + p)/4 as given by (3.29). For strong scintillations, the spectral index n is not a constant any more; it depends on S4. Fig. 1 1 shows two curves of spectral index as function of S4 obtained from the same numerical computations as in Fig. 10. We note that for the same ionospheric conditions, 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 function of frequency for different ionospheric conditions.

Although analytic solutions 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 scintillation conditions, using the phase screen approach [801-[86]. The starting point is (3.8). With the assumption of Gaussian statistics, r4 at the bottom of the irregularity slab can be computed. This can then be used as the “initial condi-

tion” for (3.57) to yield an analytical expression for the power spectrum function for the intensity on the ground [ 81 1 , [ 821

(3.59)

where

Again the phase structure function D6 appears in the expression. 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 in (3.21) and (3.26). For power4aw ionospheric irregularities of the form of (3.53) (valid for K~ < I K I I < ~ i ) , the scintillation index can be found explicitly [ 861

where J is a numerical factor dependent on the degree of anisotropy 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 function of 5 only (3.58). From (3.62), it-follows that the general scintillation index will depend on S40 in a universal manner, independent of the ionospheric condition and the propagation geometry [ 1151. The parameter tp/’ and hence S40 can be considered as the strength parameter that characterizes the level of scintillation for the ionospheric applications.

Based on (3.591, asymptotic expressions for S4 and the power spectrum for large values of 5 (or S40) have been derived for different ranges of values for p [ 8 l l , 182 l , [85 l , [ 861 . For the case p 2 4, the scintillation index is found to exceed unity for certain intermediate values of {. This is known as “focusing.” As 5 increases further, S4 approaches unity. This behavior is also found in results from numerical computations [53], [ 1721.

3) Correlation Function and Coherence Interval: The correlation fynction for the intensity scintillation is given by r4(5, r’l, r2 = 0). For weak scintillation this function can be approximated by the results from the Rytov solution (Fourier transform of (3.26)). For strong scintillation, numerical solutions of (3.54) and (3.57) give us this correlation function. Fig. 13 shows an example from such computation. Two intensity correlation functions are shown for certain ionospheric conditions. It is interesting to note the faster dropoff of the correlation at the lower frequency, corresponding to 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


D@(Tcuo) = 1 (3.64) for the multiple-scatter regime. For the power-law irregularity spectrum of the type (3.531, it can be shown that [86]

f,=5MHr

L’I00krn

r,=Wrn

Transverse C o a h t e a (m)

Fig. 13. Transverse correlation functions for the intensity of 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 is proportional to l / g . As the frequency decreases, the multiple scattering effects enter the picture and eventually dominate. These effects cause decorrelation so that the correlation distance will decrease as the frequency decreases. These two competing controlling mechanisms for the correlation of the intensity at high- and low-frequency limits will result in a maximum correlation distance occurring at some intermediate frequency. 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 [53 I . Although the results are for correlation distance, they can be transformed to those for the coherence interval corresponding to the temporal behavior of the scintillating signal. If the “frozen in” idea is valid, then the relation between correlation distance I , and coherence time 7, is simply T, = I , / 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 T, can be defmed by

(3.65)

where C is a parameter depending on the strength of the irregularity 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 the two equations in and carry out certain itera- tive solutions for the resulting differential-integral equations [581, 1 181, [ 1 191. Some asymptotic results have been obtained from the phase screen approach [ 811-[83]. The spectrum has the same high-frequency asymptote as for the weak scintillation case, but the rolloff frequency is increased, indicating a broadening of the spectrum which corresponds to the decorrelation of the signal. There is also an increase in the low-frequency content of the spectrum, 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, there have been some promising new developments using the path-integral method [ 1201 -[ 1233. The method is especially suitable for strong scintillations in the saturation region.

The discussion of any scintillation theory will not be complete if one does not mention the probability distributions of the scintillating signals. Indeed, t h i s is an area that is least developed in ionospheric scintillation theory. In the following section; a brief discussion on this subject will be given.

E. Probability Dism’butions of the Scintillating Signuls To study the probability distributions of the scintillation

signal theoretically, several approaches have been adopted in the literature. One is to use heuristic arguments to analyze the scattering process and then apply the central-limit theorem in probability study to determine the distribution. This approach has led to the prediction of joint Gaussian distributions for the real and imaginary parts of the complex signal. Application of the arguments to the Rytov solution results in the log-normal distribution for the intensity [ 1241. In the weak scintillation regime, the theoretical predictions seem to agree with the experimental data [ 1251. There have been many statistical theorems developed governing when the central limit theorem can be used. However, these theorems are difficult to apply to propagation problems [ 1261. The second approach is to theoretically calculate first the moments of the distribution and then compute the distribution. In 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 + - m3 + . . (-iwl2 ( - 1 0 ) ~ 2! 3!

(3.66)

where m, = (r” ). The probabiliy distribution function for the intensity p ( I ) can then be calculated

c -00

YEH AND LIU: RADIO WAVE SCINTILLATIONS IN THE IONOSPHERE 34 1

For this approach to work, the moments must satisfy certain convergence conditions [ 1271, [ 1281. Furthermore, the moments themselves usually are not easy to obtain. In optical scintillation problems several authors have attempted to use this approach to determine the distribution [ 1291 ; the results have not been very promising. Using the phase screen theory to compute the moments for the intensity, 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 characteristic functions [ 1301. Some results have been obtained for the optical propagation case [ 13 1 I .

To this date, these approaches and efforts notwithstanding, a satisfactory theory for the proability distributions of the various parameters of the scintillating signal has not been developed. Recently, for optical propagation problems, several authors [ 1321 have adopted a practical procedure to study this problem. This amounts to a trial and error 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 experimental data to see if the distribution yields correct predictions. Some insights can be gained from this type of investigation. As the observational data from ionospheric scintillation experiments accumulate, it will be desirable to apply this technique to study the problem of probability distributions 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 ionosphere. The presence of the earth magnetic field makes the ionosphere a magnetc-ionic medium and hence anisotropic. Fortunately, most radio frequencies used in the ionospheric scintillation experiments or in transionospheric communications are all much higher than the ionospheric 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 continuous change in relative phase between the two characteristic waves which are counter rotating and circularly polarized. Each characteristic wave will experience scattering if there are present electron density irregularities. Under the high-frequency apcroximation a stochastic wave equation for the electric field E can be derived and it shows that the characteristic waves are not coupled by the scattering process [ 1341. Making the weak and forward scatter approximation, this 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') is the normalized ith characteristic vector (circular in the present case), k(') is the propagation constant of the ith mode, and @(j) is 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 given by (3.3). Let the incident 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 fluctuating phase given by Re (@('I + &))/2 and a fluctuating amplitude 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 expressions for the variance of these fluctuations have been obtained for irregularities with Gaussian spectrum [ 1341 and power-law spectrum [ 1351. They show important depolarization effects up to 136 MHz in the ionosphere.

IV. EXPERIMENTAL RESULTS A . Irregularity Structures

We have seen from the earlier discussions that the scintillation of radio signals is intimately related to the structure of ionospheric irregularities, i.e., the space-time behavior of AN. Even when restricted to the part of the structure or the spectrum that affects transionospheric radio waves only, the spatial scales will range from submeters 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 information, a multitechnique experiment 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 density, the variance of fractional electron density fluctuations, and the irregularity spectrum. Only after possessing such information on a global basis can one attempt to construct a global scintillation model [ 1391. We review briefly such information 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 to be 1000 km in the north-south direction [ 1431. This north-south size is comparable to the airglow meaurements made recently, which are indicative of regions of depleted electrons or bubbles [ 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 kilometers [ 1461 -[ 1481. In all geographic regions, the nighttime irregularities that produce scintillations are found to be mostly embedded in the F region ionosphere from about 200 km to 1000 km [138], [145], [149]-[151], but daytime scintillations are caused mainly by E region irregularities [ 1491, [ 1521, [ 1531. The thickness of the patch is found to vary from tens of kilometers to hundreds of kilometers [ 1381,


1141, [371, [1541, [1551. There is some evidence, at least at temperate latitudes, that the fractional fluctuation of electron 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 convenience [231, [ 101 ] . The first suggestion that the spectrum might follow a power-law form came from satellite scintillation data [40], [4 1 ]. Making use of the phase screen scintillation theory, it was then deduced that the implied irregularity spectrum had a power-law form given by (2.8) [42]. At about the same time the electron density data measured by the probe on board the satellite OG06 became available and a power-law spectrum was also obtained [43], [44], [ 1571. However, these in-situ measurements limited the spectrum given by (2.8) to a two-decade range from K = lo-' m-l (scale size about 7 km) to K = lo-' m-l (scale size about 70 m). The ground scintillation observations also had difficulties in extending the spectrum because of data noise and the well-known Fresnel filtering effect discussed in Section I11 [ 1581. Yeh et al. [ 531 then suggested to combine the scintillation or in-situ data with the radar backscatter data for the purpose of extending the spectrum. This is useful because a radar senses the irregularity spectrum at one-half of its operating wavelength when con- figured in the backscatter mode. As a matter of fact using the published radar data at 50 MHz for the equatorial spread-F [36] , a preliminary calculation showed the power-law could be extended t o K = 2 m-' (scale size down to,3 m) [531. With more careful calibration of the radar and the use of coordinated scintillation data, the radar returns were found to be more than 40 dB less than that expected, based on the extrap olated power law down to 3 m [54]. Therefore, these authors suggested a Gaussian cutoff of the power-law spectrum near the 0' ion gyro-radius which they took to be 3.35 m. Such a cutoff was found to be inconsistent, at least sometimes, with the later observed radar returns at 1 m and 36 cm [ 1591- [161], and even as small as 11 cm [551. Even though the precise nature of the spectrum down to such a small scale (not much larger than the Debye length of 6.9 cm as estimated in [55]) is not known, its generation must require a sequence of plasma instability processes peculiar to the equatorial geometry and environment. Coordinated experiments showed that the 3-m radar scatter would usually disappear around or later than local midnight even when the UHF and L-band scintillations were very strong [ 138 I . This suggested the possible change of the inner scale with time. Furthermore, more careful analysis of both the scintillation data and the in-situ data gave evidence that even the spectral index itself evolved with the irregularity intensity [ 1621. These data showed a power-law index de- creasing with the strength of phase scintillation. There was also evidence indicating that the equatorial irregularities may have a two-component power-law spectrum with a higher spectral index for smaller irregularities [ 2 11 .

As to the three-dimensional nature of the irregularities, the f i t suggestion for magnetic field alignment was made by Spencer [ 1631. Early reports of experimental measurements on the size showed an axial ratio of 4 through 8 to 1 as r e viewed by Herman [32] and the more recent results using a large array [ 1561. The enhancement of scintillations when the

orbiting satellite moved within the same L-shell as the observational station led Rino and his colleague [ 1641, [ 1651 to suggest sheet-like irregularities in the auroral zone.

B. Signal Statistics As discussed in Section 111-E, there does not exist at present

a satisfactory and rigorous theory which predicts the probability distribution function of a scintillation signal. It is only known that when the scintillation is well in the saturation regime, the amplitude distribution approaches that given by Rayleigh [ 12 1 1 . Several heuristic arguments have been proposed that lead to simple distributions as working models. Over the years, four models have gained popularity among investigators in the field. The first is based on the phase- changing screen idea plus some assumptions about the statistics of electron density irregularities. This model postulates that the phase undergoes a random walk in a phase-changing screen and becomes a Gaussian process [ 671. When this idea is extended by using the Rytov solution, it is natural t o s u p pose that the logarithmic amplitude and the phase are jointly Gaussian [ 1321, [ 1661. Under this postulate, the intensity has a log-normal distribution, and the phase has a normal distribution. If, instead of logarithmic signals, similar assump tions are applied to the quadrature components themselves, the in-phase and quadrature components become jointly Gaussian. This is our second model. In this case the general expressions for the intensity and phase distributions are complicated, but they lead naturally to Rician statistics and Rayleigh statistics in limiting conditions [ 921, [ 1671. The third model is an approximation to Gaussian stastistics and is known as Naka- gami-m distribution for intensity [ 1681, given by

The parameter m is equal to the inverse square of the scintillation index S4 [ 941. The recent realization that the irregularity spectrum falls slowly in a power-law form with the wavenumber suggests the coexistence of large irregularities that produce only refractional effects and small irregularities that produce only diffractional effects. This suggestion led Fremouw e t 171. [ 1691 to propose a two-component model, which decomposes multiplicatively the complex signal into a slow refractive component and a fast diffractive component. The diffraction component is postulated to obey generalized Gaussian statistics, and the refractive component obeys log-normal statistics. In all these four models, the intensity and phase distributions are uniquely defined by the second-order moments of the received signals, which can be computed easily from the observed scintillation data. The distributions so obtained in each case can be subjected to chi-square tests against histograms of the observed intensity and phase. This procedure has been carried out by Fremouw et al . [ 1671 who find that the normal distribution has the overall best fit for the phase and the Nakagami-m distribution has the overall best fit for the intensity. The amplitude (or intensity) distributions of scintillating radio signals have been studied by many investigators [ 1251, [ 1701, [ 171 ] and the Nakagami-m distributions have been found to represent a good fit for the intensity by Whitney et al . [ 171 1 . This is further demonstrated in Fig. 15 taken from Rino [ 1721. But the results of Fremouw et al. [ 1671 suggest that a suitable bivariate distribution would be one that reduces t o the product of the Nakagami-m distribution for the intensity

YEH AND LIU: RADIO WAVE SCINTILLATIONS IN THE IONOSPHERE 34 3

1 k O . 3 0 , li"$, , 1 2 1 2 3 1 2 3 1 2 3

ANCON, VHF, DAY 0 6 3 , 0349 UT

u KWAI. L-BAND. KWAJ. UHF, ANCON, VHF. DAY 104. 1218 UT DAV 0 8 4 , 1306 UT DAY 08B. 0346 UT - - ". -

2 3

RELATIVE INTENSITY

ANCON, L-BAND, DAY 089.0415 UT

ANCON, VHF, DAY 056.MZO UT

1 2 3 1 2 3 1 2 3 4 5 6 1 2 3 4 5 6 7

RELATIVE INTENSllY

Fig. 15 . Intensity scintillation probability distribution at various scintillation levels and the corresponding Nakagami distributions (solid curve). (After Rino [ 1721 .)

and the normal distribution for the phase in the limit of zero correlation between the two variates.

C. Scintillation Index Experimentally the scintillation index is conveniently used

as an indicator of scintillation strength. In the literature there are a number of such indices in general use. These indices of course can be converted and related to each other empirically or through a given probability distribution [ 941. For quantitative work, the scintillation index S4 defined by (3.27) has been found to be most useful. Under weak scintillation conditions, the scintillation index S40 has a f-" frequency dependence with n = ( p + 2)/4 as given by (3.29). In the ionosphere the spectral index p is found to be 4 * 1 [42], which implies n = 1.5 f 0.25. Since experimentally S4 indices are measured at discrete frequencies, t o test the frequency dependence between any two pairs of frequencies, say fl and f z , one can define [ 173 I

When n is independent of frequency, (4.2) is consistent with the frequency scaling law of S4 af-" as desired. Using the 40-MHz and 140-MHz data recorded at Boulder, the scintillation index n(40/140) has been obtained and is plotted against s4(140) in Fig. 16 [ 1741. The behavior of the data points shows that for weak scintillations, the spectral index n a p proaches a value of 1.6, which implies a p value of 4.4. This is in agreement with the in-situ data. However, as the scintillation index S4( 140) increases, the data points of Fig. 16 clearly depict the dependence of n on S4, suggesting the dependence of n on f. The general trend, indicated by the solid curve, shows the weaker dependence of n on f as S4 increases. This is in agreement with the numerical results [ 531. The weakened dependence of n on f is caused by multiple scattering. To take multiple scattering into account one needs to solve the r4 equation (3.48) and relate F4 to S4 through (3.58). Unfortu- nately, even though some numerical and asymptotic solutions

1 .

L

00 0.1 02 03 a4 05 S I 0 9 WO MHz

Fig. 16. The dependence of spectral index n(140/40) defined by (4.2) as a function of S, (140). The experimental data were recorded at Boulder, CO, from radio transmission of the geostationary satellite ATS-6. (After Yeh and Liu [ 1741 .)

of F4 do exist in the literature, there is no analytical solution. What can be done is to realize that the r4 equation can be put into a dimensionless form (3.54) by proper normalization, in which the only parameter is the spectral index p . As discussed in Section 111-D2, we would expect a universal solution parameterized in p but independent of geometry and irregularity strength. Therefore, in multifrequency observations, one can assume that S4 computed or the highest frequency satisfies the Rytov solution and hence follows the frequency scaling law f-", n = ( p + 2)/4. For other frequencies, the observed S4 is organized according to S40 based on the frequency scaling law f - (p+2)/4 from the highest frequency. This has been done by Rino and Liu [ 11 51, with their results shown in Fig. 17. In spite of diversity of observational locations and the limited data points, the general trend of linear increase for small S40 and asymptotic approach to unity for large S40 is clear. There are also distinct differences. The equatorial data from Kwajalein and Ancon approach unity S4 from below whereas both the temperate latitude data from Boulder and high latitude data from Poker Flat show focusing

344

1

1

s4

0

1 1 1 1 1 1 0

0 +

C A

+ + + -

:+ mY A

$ I

WIDEBAND - + KWAJALEIN A POKER FLAT

ANCON

ATS-6 0 BOULDER

I I I 1 2 3 4 5 6

s40 Fig. 17. Multistation and multisatellite S, scintillation data plotted

against S, . (After Rino and Liu [ 11 5 1 .)

'Of 0 PHASE SCINTILLATION

t

10.2 I I to2 10'

FRECUENCY (MHz 1

Fig. 18. Standard deviations of phase scintillations as a function of frequency from Wideband Satellite data. The curves are average values over many passes. (After Quinn [ 1781.)

near S40 values of approximately 1.5 where the scintillation index S4 exceeds unity.

In Section 111-C2 the variance of the phase fluctuation (Si) under the Rytov solution was shown to vary as l / f2 . Fig. 18 shows the standard deviation of the phase fluctuation from the Wideband Satellite data [ 178 1 . The l/fdependence is apparent in general except for the Ancon data. Certain saturation tendency is seen in this case at lower frequencies. This may be due to multiple scattering effects in the stronger scintillations at Ancon.

Global behavior of scintillation morphology and modeling have been investigated and studied because of their importance to geophysics and their applications [ 16 1 , [ 1751, [ 176 1 . This subject is being reviewed in the companion paper [25], and hence will not be reviewed here.

D. Spatial Signal Characteristics The spatial signal characteristics are essentially described by

the one-frequency two-position mutual coherence function r2

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

discussed in Section 111-Dl and the intensity correlation function r4(3', il, i2 = 0) discussed in Section 111-D3. These functions are interesting in their own right because of their applications to spatial diversity. Under the condition of frozen flow the spatial fluctuations can be converted to temporal fluctuations. Therefore, these functions can also be taken to describe the fading rate as observed by a fmed observer. The rate of fading has implied impact on communications.

Section 111-D3 describes the behavior of intensity correlation distance as being controlled by the two competing factors: Fresnel filtering and multiple scattering. As a function of frequency, this correlation distance behaves as those shown in Fig. 14. To check this behavior experimentally, one can make use of multifrequency scintillation data. One such study, made by Umeki et al. [ 1731, uses radio transmissions of the geostationary ATS-6 at 40, 140, and 360 MHz. First the intensity correlation functions are computed from the multifrequency data under various scintillation conditions. The correlation times are then scaled from the correlation curves by noting the time required for the normalized correlation function to be reduced to 4. To avoid dependence on the ionospheric drift velocity u o , the correlation times in each set at lower frequencies are further normalized by the correlation time at 360 MHz. All data points thus obtained are then organized by using S4 at 360 MHz, and the average behavior within each S4 range is indicated by curves shown in Fig. 19. For a weak scintillation case, as indicated by curve 1, the Correlation interval follows closely the f-l12 behavior. As the scintillation level increases, such as curve 2, the frequency dependence departs from f-l12 behavior. When the scintillation level is fairly strong, such as curve 3, the correlation shows a peak near 140 MHz. In this way, an experimental verification of the theoretical curve depicted in Fig. 14 is obtained.

In Section 111-D3, an expression for the coherence time based on phase-screen theory was given in (3.63). Experimen- tal verification of the formula has been demonstrated by Rino and Owen [ 1771 using data from the DNA Wideband Satellite.

For phase fluctuations, the behavior of the correlation time is quite different. Fig. 20 shows the mean normalized correlation time TO derived from phase correlation functions using multifrequency data from the Wideband Satellite [ 1781. The data were grouped into three categories according to the value of scintillation index S4 at L-band (1239 MHz). Those sets of data with &(L) > 0.3 are the first group. The second group corresponds to 0.1 < S4(L) < 0.3 and the third group to S4(L) < 0.1. We note that for weak scintillation, as in group 3, T+ is almost a constant across the whole frequency band. This is due to the fact that the Fresnel filtering is absent in the phase scintillation, and therefore, no f-Y2 dependence is present. For strong scintillations in groups 1 and 2, however, there are some indications of decorrelation of the phase fluctuation due to multiple scattering at the VHF frequencies.

In the statistical characterization of the signal, power spectra have been found to be useful because of their intimate relation to the irregularity spectrum. For the received signal, the spectrum can be computed for the amplitude, the phase, and the two quadrature components. Each of these will be described in turn in the following.

In case of weak scintillations, the theory is well developed as discussed in Sections 111-B and 111-C. The spectrum of the amplitude is nearly flat up to a rolloff frequency known as the Fresnel frequency C ~ F , above which the spectrum falls with a power-law dependence S2-1-p. Phase, on the other hand, is


2.5

' I I I

RMS S ~ E A D i RMS SPREAD S, at 360MHz ot 40 MHz I at I40 MHz ii(140/40)

c0.02 f 13% 1.6 1 +20?4 - IJB 1 2 I I % f 9% 0 93 + 25% 214%

0.06,0.13 a91 i ?

r H I'4 z \

O 5 1 40 I40 360 FREQUENCY (MHz)

Fig. 19. Normalized intensity correlation intervals as a function of frequency for four scintillation conditions. (After Umeki er al. [ 1731 .)

2 r - - - - - l t 4

GROUP I

dominated by the low-frequency components, and its spectrum is of the form !22- ' -p down to the lowest frequency allowed by detrending of the data. An example of the 4@MHz amplitude and phase spectra is shown in Fig. 21 [ 501. The fact that the phase is dominated by low-frequency compe nents poses a practical problem, viz., the results on statistical analysis of the phase data depend on the cutoff frequency of the detrending process as found by several authors [ 501, [ 901.

Since our understanding of the spectrum under weak scintillation conditions is nearly complete, it is possible t o compare with the theoretically computed spectrum by adopting a model irregular ionosphere. This has been done [ 5 1 I . The degree of agreement between the experimentally observed spectra of a 40- and 140-MHz amplitude and the corresponding spectra computed based on the theory are depicted in Fig. 22. The agreement is striking. Furthermore, the S4 indices for the data are 0.54 rf. 0.04 at 40 MHz and 0.076 rf. 0.006 at 140 MHz, while the corresponding theoretical S4 indices are 0.59 and 0.077, respectively. This comparison again shows good agreement. The spectra at both frequencies show a high- frequency asymptote with a dependence which yields p = 4.5, a value close to 4.4 deduced from the consideration of the radio-frequency dependence of S4 index discussed in the last section. Additionally, the experimentally observed rolloff frequencies are 0.025 Hz at 40 MHz, 0.045 Hz at 140 MHz, and 0.07 Hz at 360 MHz which follow closely the theoretical f 1 I 2 dependence. When the spectra are normalized by their respective rolloff frequency, the spectral curves on these three radio frequencies fall very closely on top of each other as shown in Fig. 23 by Umeki et al. [511. These discussions demonstrate not only the agreement between the theory and experiment but also the internal consistency of the results.

When the scintillation level increases to the saturation regime,

\L’ =

3.3

SET

1A 4

0 PH

ASE

AMPL

ITUD

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ATA

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Pow

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and

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ms p

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is 1.

628

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he cu

toff

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of

the

det

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roce

ss is

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03 H

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305

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11 .)


c

c 3 -10

Z L

- 30 F S E T 1 NORMALIZED

IH+ 40 MHz

140 MHz

360 MHz -

-40 1 I 90% CONFIDENCE LIMITS

I- -I

FREQUENCY Fig. 23. Power spectra BS a function of normalized frequency n / n ~ ,

where S2p is the rolloff frequency or the Fresnel frequency. The S, scintillation indices are 0.54 at 4 0 MHz, 0.076 at 140 MHz, and 0.016 at 360 MHz. The data set is identical to that shown in Fig. 22. (After Umekieral. [Sl].)

the theory is on a weaker foundation. To investigate the saturation effects on spectra, a different data set is selected [ 5 1 1 . The results are depicted in Fig. 24. The three spectra are again normalized by the corresponding Fresnel frequency as in Fig. 23. A comparison of the two figures shows that for strong scintillations the rolloff is more gradual and the spectrum extends out to high frequencies. This increased high- frequency content during severe scintillations is entirely consistent with the decreased correlation interval discussed earlier. Even though the data at three frequencies have different scintillation indices, the high-frequency asymptotes in all three spectra have very closely the power-law dependence of Using a phasescreen model, Rumsy [82] and Marians [ 831 find that the high-frequency asymptote is independent of the scintillation level, in agreement with our experimental results. Other numerical and asymptotic results indicate that the intensity spectrum first rolls off very rapidly and then develops the power-law tail [ 1 161. However, a more complete theory valid in the multiple scatter regime is not yet available at present time.

The decomposition of a complex signal into amplitude and phase is certainly useful. For some applications, however, it is more convenient to decompose the s i g n a l into quadrature components defined by

X = A cos4 Y = A s i n @ (4.3)

where A = A o + A l is the detrended amplitude with the mean A . restored, and 4 is the detrended phase. Here X is referred to as the in-phase component and Y the phase-quadrature component. For weak scintillations, 4 is small and thus (4.3) can be expanded to the second order, with the fluctuating components given by

x1 = A 1 -A042/2 Y1 = A o 4 + A , 4 . (4.4)

Equation (4.4) indicates that, for weak scintillations, the statistics of the in-phase component X1 should be very similar t o those of the .amplitude A l , and the statistics of the phasequadrature component Y l should be very similar to those of the phase 4. These notions are born out by data plotted in Fig. 25, which shows similarity between the amplitude and the &-phase component, and Fig. 26, which shows similarity between the phase and phasequadrature component. How- ever, situations change as scintillation levels increase. In the saturation regime, the amplitude fluctuation is fast, due to the decorrelation effect discussed earlier, and the phase fluctuation is large but slow because of domination by large irregularities. This combination of amplitude and phase fluctuations renders the power to be divided nearly equally in the quadrature components. Support for this observation can be found in Fig. 27, where the spectra for two quadrature components nearly lay on top of each other. Additionally, the quadrature components agree with the amplitude spectrum, as depicted


30 I I I I I I l l I I I I I l l 1 I I 1 I I I l l

SET 3 NORMALIZED - 40 MHz S4= 1.42

* 140 MHz S4.054

- 360 MHz S4 = 0.13

1 90% CONFIDENCE LIMITS

FREQUENCY Fig. 24. Power spectra as a function of normalized frequency 52152F.

The S, scintillation index for each of three radio frequencies is shown in the legend. The ATS-6 data were recorded on January 7, 1975, 0700-0750 UT. (After Umeki e t al. [S 11 .)

in Fig. 28. These results suggest that, under intense scintillation conditions, the power spectra of the two quadrature components are controlled by the fast fluctuating amplitude. The equalization of power between the two quadrature components comes about because of the slow but large phase fluctuations [SO].

E. The Mean Field and the Coherence Function In Section 111-D, it was shown that the mean field received

( u ) =A,-, exp [ - r ~ h Z L A ~ N ( 0 ) / 2 ] . (3.41)

The multifrequency Wideband Satellite data provide us the mean field ( u ) at different frequencies. From the values of ( u ) , the parameter C%N = LA*N(O) for a given satellite pass can be determined. From the theory, this quantity should be a constant independent of frequency for a given satellite pass. It is only dependent on the propagation geometry and the irregularities. Table I gives some examples from the Wideband data [178], [179].

We note that for these examples, the- strength parameter C i N is approximately constant for the same pass. Using these parameters and geometric optics, the rms fluctuations of phase us have been computed. They compare quite nicely with the observed US.

It should be mentioned that the data often showed mean field ( u ) with significant imaginary parts, especially for VHF signals at 137 MHz contrary to the theoretical prediction. Several possible causes may contribute to this discrepancy:

on the ground is given by

TABLE I IONOSPHERIC SCINTILLATION STRENGTH PARAMETERS c ~ N DERIVED FROM

WIDEBAND SATELLITE DATA AND COMPARISON OF OBSERVED AND COMPUTED PHASE FUNCTIONS

Pass Freq. ( H H ~ ) ~ ~ ~ 2 ~ 1 0 - ~ ~ cs (rad) computed a

K2R4" 378 2.06 0.984 1.02

447 .1.90 0.854 0.86

1239 1.35 0.241 0.3

mean 1.76

K4 " 378 1.43 0.59 0.85

447 1.46 0.59 0.72

1239 1.13 0.15 0.25

mean 1.34

P3 378 0.59 0.54 0.55

447 0.59 0.46 0.47

1239 0.43 0.14 0.17

mean 0.54

P3R16 378 2.75 1.11 1.17

447 2.67 0.94 0.99

1239 1.75 0.29 0.35

mean 2.39

aK2R4 and K4 are passes over Kwajalein. bP3 and P3R 16 are passes over Poker Flat.


0 0

d m

0

9 0 N

0 0

0

0 0

md 0

Eg

57

5s

WO

c.

ad =?

0 0

0 m I

0 0

0 = I

0 0

d YI 1

PBHER SPECTRR. SET I V 1 I I O M H Z

A M P L I T U D E

+ IN-PHASE COMPONENT

90% C O N F I D E N C E L I M I T S

. - , . . . . . . . . 1. E-2

. . . . . . . . F R E Q U E N C Y IN HZ l .E-1 1.bo

Fig. 25. Comparison of amplitude and in-phase component power spectra fy data recorded at 140 MHz, Au- gust 25, 1974, 0525-0548 UT. The S, scintillation index is 0.522 and the r m s phase fluctuation is 1.397. (After Myers et al. [ 501 .)

the assumption of forward scattering may be violated; the spherical nature of the incident wave may become important and uncertainties in measuring the phase may play a role.

The coherence function r 2 ( z , p') is another function the exact solution of which can be obtained from theory (Section 111-Dl ). Experimental verification of this theoretical result has not been as successful as for other cases. Some prelimi- naryexamplescanbefoundin[1781,[1791.

V. TEMFQRAL BEHAVIOR A . Average Pulse Intensity and Two-Frequency Mutual Coherence Function

The theoretical and experimental discussion in the previous chapters on the ionospheric scintillation phenomenon have been mainly for monochromatic signals. In many situations, especially for communication applications, and lately for the interstellar scintillation of pulsars [ 1801 -[ 1821, the evolution of the temporal behavior of a signal as it propagates through the irregularities is of interest. Consider a pulsed signal with a carrier frequency w, and frequency spectrum f(o) incident on the ionosphere irregularity slab (Fig. 7). The average pulse intensity received on the ground can be shown to be given by [ I831

(l(t))=IIF*(i12fF(i11)r1,,(i11,i12)

+-

-00

. exp [i(ill - i 1 2 ) t - ( k , - k 2 ) z 1 d i l l di12 (5.1)

where i l i = w . - w,, F ( n ) = f ( w , + ai), and ki = koi [ 1 - 4 m e N o / k ~ i l 'I4, koi = w i / c . rl, is the two-frequency one- position mutual coherence function defined by (3 .38) , satisfy- ing (3 .42) . Note that if the electron density variation of the background ionosphere N o ( z ) is to be taken into account, the term (k, - k 2 ) z should be replaced by I:(k, - k 2 ) d { .

From (5 .1) , we see that the two-frequency mutual coherence function rl, , plays the important role in describing the propagation of the average intensity in a random medium. It has been measured experimentally [ 2121. Indeed, a coherence bandwidth A i l , can be defined as the value of A i l = w2 - w1 at which rl,, reduces to l / e (or 3) of its maximum value at A i l = 0 [ 1821, [ 185 1. Comparison of this coherence bandwidth with the bandwidth of the signal f ( w ) will indicate how much the averaged pulse will be distorted. For the Rytov solution of rl, , (3 .45) , this coherence bandwidth is approximately given by [ 1081

(5 .2)

where @o is given by (3 .13) and

32' - 3 z L + L 2 A4 4 kz A0

d t = - ( 5 . 3 )

Here, kc = w , / c , A0 , and A4 are the expansion coefficients for the transverse correlation function A A N for the electron density fluctuations such as the one given in (2 .21) . We have adopted an expansion


POHER SPECTRR. SET I V lUOMHZ

- PHASE

-8- PHASE QUADRRTURE CBMPONENT

$ 9OZ CONFIDENCE L I M I T S

0

1. E-2 FREQUENCY I N HZ

1. E-1 1.00

Fig. 26. Comparison of phase and phasequadrature component power spectra for the same data as those shown in Fig. 25. (After Myers er a2. [ S O ] .)

= A o + A2p2 + A4p4 + * . (5.4) tions

On the other hand, under extremely strong scintillation conditions such that the transverse correlation distance of the field becomes small as compared with that of the irregularities, asymptotic solutions for rl. can be obtained [ 1091, [ 1861, . {a" [F(Stl)rl,l e - j k l z ~ / a ~ : } n , = n , d n 2 . (5.6)

M(")(z) = 2n(j 1" F * ( s ~ , ) ejkzz l: [ 1871. These asymptotic .expressions can then be used in

in general, rl, l cannot be obtained analytically. Instead, (3.42) has been solved numerically as discussed in Section 111-Dl. However, these numerical solutions are not easily a p plicable in (5.1) to obtain the average pulse intensity.

In many applications, the exact shape of the pulse may not

. - In order to interpret the temporal moments in terms of physi-

condition M(O)(z) = 1 which normalizes the energy of received signal t o unity and ii) time origin condition M(')(O) = 0 which sets the time origin at the center of the symmetric envelope of the incident signal. The normalization condition requires

(5*1) to compute the average Pulse cal we impose two conditions: i)

be needed. Rather, rough descriptions such as the time of arrival and the pulse broadening caused by the random scatter- F*(St ) F(S t ) rl, (St, St) dS2 = 1 (5.7) ing are important. This leads naturally to the idea of the temporal moments. and the time origin condition implies

P +- (5.8)

(5*5) We note from (5.6) that for the temporal moments, the complete expression for the tw6frequency mutual coherence func-

Substituting (5.1) into (5.5), we obtain after some manipula- tion is not needed. Instead rl, and its derivatives evaluated


POWER SPECTRA. SET 'I1 UOHHZ

IN-PHRSE CBHPONENT

* PHRSE OURDRRTURE COMPONENT

3 90% CONFIDENCE LIMITS

1. E-2 FREQUENCY I N Hi! 1.E-1 1.00

Fig. 27. Comparison of power spectra of two quadrature components of signals at 40 MHz, August 23, 1974, 0505-0558. The scintillation index is 1.30 and the rms phase fluctuation is 2.677 rad. (After Myers eral. [ S O ] . )

at a, = sZz are the quantities that are required in evaluating torted pulse. To see how tne rl(f)l's can be solved analytically, M ( " ) . This suggests the expansion of rl, in terms of the we again specialize in the case of plane wave incidence. For variable 6 = ( k z - k l ) / k z this case (3.42) becomes

and If we let

(5.10)

The temporal moments M(") depend only on rl:], I'!:!, . * , rp). It turns out that although r l , l cannot be obtained analytically in general, its derivatives evaluated at 6 = 0, the r,cf)l's, can be solved exactly, independent of the strength of the irregularities.

The physical interpretation of the first few temporal moments are quite obvious. M(')(z) can be interpreted as the mean arrival time of the signal at position z. The second moment is related to the mean-square pulsewidth since M ( 2 ) - [M( ' ) ] ' measures the broadening of the pulse. The higher order moments are related to the skewness, etc., of the dis-

then from (5.11) we obtain

(5.13)

At z = 0, F l , l = 1 so that W = 1. Equation (5.13) is then solved for W by letting

W = W o + w 1 6 + W z 6 ~ + - ~ * . (5.14)

A hierarchy of equation for the Wi's is obtained from (5.13)

LW, =gn, n = 0 , 1 , 2 , * . * (5.15)

352 PROCEEDINGS OF THE BEE, VOL. 70, NO. 4, APRIL 1982

POUER SPECTRA. SET I 1 UOMHZ

AHPLITUOE

IN-PHASE COMPONENT

$ 90% CONFIDENCE LIMITS

I i FREOUENCY IN HZ

1 .E-2 l.E-1 1 .oo Fig. 28. Same data as Fig. 27 but for amplitude and in-phase component. (After Myers e t aZ. [ S O ] .)

where L is an operator of the form

(5.16)

and

go = o

(5.17)

Equation (5.15) can be solved in a straightforward manner. With the initial conditions WO = 1, W1 = W 2 = - . = O a t z = O , we obtain

(5.18)

and

P Z . I

w n = w~ .Io w o w -gn(f) df, n = 1 , 2 , . * * . (5.19)

Equations (5.18) and (5.19) together with (5.12) determine the rl, 's for z < L . These functions evaluated. at z = L can then be used as initial conditions to solve for I'{:i's below the irregularity slab, which also satisfy a hierarchy of equations

derived from the equation

a I ' 1 , 1 -- -i - a Z 2(1 - 6 ) k 2

v 2 r l , l , z > L . (5.20)

Therefore, a procedure has been developed which enables us to obtain the derivatives of rl, 1, the I'ifik, for any strength of the irregularities. These derivatives can then be used to compute the temporal moments. Although the procedure described applies only to plane wave incidence, it can be generalized to the case of beam waves [ 1891.

In terms of the temporal moments, a mean arrival time of the pulse can be defined by

tu (z) = M(')(Z)/M(O). (5.21)

For ionospheric application, it can be shown that [ 1901, [1911

ta (z) = z/c + .If,' + plf: + - * (5.22)

where the fmt term is the free space travel time at the speed of light c. The second term is the excess time delay due to the total electron content (TEC) of the ionosphere and is inversely proportional to the square of frequency. a is given by

where f, = (e2No/4a2rneo)1 /2 is the plasma frequency of the background ionosphere.

The third term in (5.22) is inversely proportional to the

YEH AND L N : RADIO WAVE SCINTILLATIONS IN THE IONOSPHERE 353

0'

\ 1 0 0 200 5oom

Frequency ( MHz 1 Fig. 29. Excess distance as a function of frequency for a Chapman

ionospheric layer with N o = e l / n ~ - ~ , H = 100 km, h, = 300 km. The ionospheric irregularities are assumed to have rms fluctuations of 100 percent from the mean, an inner scale of 10 m, an outer scale of 5Okm,andasignalbandwidthoflOMHz. (AfterYehandLiu[191].)

fourth power of the signal carrier frequency. It is given by

+ 1' f,'(z - 5 ) d f (5.24)

where A,, is the expansion coefficient of the transverse correlation function A , of AN/No

A , @ ) = A r o + A r 2 p 2 +Ar4p4 + - . - . (5.25)

We note that instead of the correlation function A A N for electron density fluctuation A N ( ; ) , the correlation function for the percentage electron density fluctuation AN/No is used here. This is due to the fact that when the background density variation is taken into account, the random field A N / N o , rather than A N itself, is more likely to be homogeneous [ 1921. Therefore, in deriving (5.24), A,, instead of A A ~ , appears.

The first term in (5.24) is due to higher order dispersion of the ionospheric plasma; the second term is due to the finite bandwidth v2 of the pulse. The third term is due to random scattering. In cases of severe ionospheric scintillations, the third term dominates the 1 / f : contribution to the excess time delay. Fig. 29 plots the l/f: and l/f: contributions to the ranging errors caused by the excess delay in arrival time for certain ionospheric conditions [ 19 1 1.

Another quantity of interest is the meansquare pulsewidth defined by

r Z ( z ) = M(2)(z)/M(O)(z) - t ( : (z) . (5.26)

For ionospheric applications, it is given by three terms [ 1901

where

(5.28)

Here ~ ( 0 ) = To is the original pulsewidth, T~ represents broadening due to dispersion, and r2 represents broadening con- tributed by scattering Some numerical examples for the pulse broadening effects in the ionosphere are given in [ 191 1 .

Moments higher than the second have also been obtained 11931. The third moment is related to the skewness of the pulse and the f o a moment is related to the kurtosis. These higher moments can be used to estimate the stretching of pulses due to propagation effects. These estimates have obvious implications in digital communications [ 19 1 ] .

C. Construction of the Mean Pulse Shape from the Temporal Moments

When the temporal moments M(") are known, it is possibIe to reconstruct the mean pulse envelope ( I ( t )> by applying the technique of orthogonal polynomial expansion [ 1941.

Let us consider a set of orthogonal,polynomials def ied by

$ ? J n ( t ) = 2 s k t k , n=0,1,2; . . ;o l , ,#O. k =O

(5.29)

These polynomials satisfy the orthogonal relation

J- 9, ( t ) w( t ) 9, ( t ) d t = 0, m f n (5.30)

where ~ ( t ) is the weighting function associated with the given set of polynomials. Depending on the forms of the weighting functions, different orthogonal polynomials can be obtained. Among the most commonly used ones are the Legendre, Laguere, Chebyshev, and Hermite polynomials [ 1951.

To reconstruct the pulse shape, define an approximate average pulse by a series

N (5.31)

This approximation is based on the criterion that the weighted meansquare error of the approximation from the exact average pulse defined by

is minimized. From (5.30) and (5.31), it can be shown that this minimization condition leads to the following determina- tion of the coefficients c, :

lW ( I ( t ) ) 4, ( t ) d t

1; w(t ) 4; d t

c, = (5.33)

Substituting (5.29) into (5.33) and making use of the definition for the temporal moments, we obtain


(5.34)

The expansion coefficients c,, in the expression for the approximate pulse shape are now related to the temporal moments A d k ) , k = 0, 1, * a , n. Therefore, knowing the first N temporal moments, we can COnstNCt the polynomial expansion I N ( t ) as an approximation to the average intensity of the pulse ( I ( t ) ) in the least square sense defined by (5.32). For a given value of N , the error in the approximation can be estimated [ 1951.

The choice of the weighting function and the corresponding polynomials seems to be quite arbitrary. In practice, however, one can start from the original incident pulse. The weighting function is chosen as a nonnegative function that can be made to approximate the intensity function of the original signal closely. The corresponding orthogonal polynomials can be found using the Schmidt orthogonalization procedure [ 195 I . In [ 1941, this procedure was used to construct the average pulse shape of an original Gaussian pulse as it propagates through a random medium.

D. Numerical Experiments Although it is common knowledge that temporal radio sig-

nals are usually degraded after propagating through the ionosphere under the scintillation environment, there are very few precise data sets against which the theoretical predictions can be checked. However, there are a number of simulation studies carried out on computers which attempt to check the theory, t o enhance our physical understanding of the problem, or to elucidate the phenomenon. As described in the previous two sections, both amplitude and phase of a sinusoidal wave will experience fluctuations when there exists scattering from ionospheric irregularities. For propagation of temporal signals, the important point is how the amplitude and phase vary across the signal bandwidth. Let A (0) be the amplitude function and $(a) be the phase function. Then it can be shown that the time delay in the arrival of the pulse is related to the slope of the phase function at the carrier frequency, the pulse spreading is related to the curvature of the amplitude function at the carrier frequency, and the skewness is related to the third derivative of the phase function at the carrier frequency [ 1961, [ 1971. For a particular realization of the turbulent ionosphere corresponding to an equatorial bubble, A(w) and #(a) may have the behavior shown in Fig. 30 which is obtained numerically [ 1971. Substantial distortions in A and $ can be seen across the frequency band above the 500-MHz carrier frequency. These distortions in A and # are responsible for the instantaneous deviations in time delay, pulsewidth, and skewness. As the-irregularities drift such as in a frozen flow, the time delay, the pulsewidth, and the skewness will fluc- tuate with time. The mean values of these quantities should agree with those derived earlier by using the temporal moments ((5.22) for mean arrival time and (5.27) for meansquare pulsewidth). Such simulation studies have been made and good agreement with the theory has been found [ 1981. For spread spectrum systems using a direct sequence PN scheme the received degraded signal plus noise is correlated with an approximately delayed PN modulation waveform. The correlator output under three scintillation levels may

l29 1.6 i I .4

.6

.4

0 . .2676 .3376 .3676 -4376 4876 .5376 .5676 .6376 .6876

.3126 ,3626 .41M .4626 .5126 .5626 .6126 .6626

1 q. t

5 '-h E 6 .

I . t 0 . 1 . . . : . . : : : : : : . : . .

.2876 .3376 .3876 .4378 .4876 .6576 . S O 7 8 .6378 . E 0 7 6 .3 176 . 3676 .4 126 .4628 a 6 128 .6826 .6 128 .E626

FREOUENCY IN GI42 (b)

Fig. 30 The amplitude (a) and the phase (b) of the transfer function aaoss the frequency band centered at a 500-MHz carrier frequency. The ionosphere is simulated from the in-situ data corresponding to an equatorial bubble. (After Tucker (1971 .)

behave in a manner shown in Fig. 3 1. When the coherence bandwidth is comparable to or greater than the signal bandwidth (top frame of Fig. 31), the shape of the correlator output suffers little distortion even though the peak value fades up and down with time. With the increased scintillation level, the coherence bandwidth shr inks t o one-fourth (middle frame of Fig. 31) or one-tenth (bottom frame of Fig. 31) of the signal bandwidth; severe distortions in the correlator output are possible [ 1991. Such distortions may pose difficult problems to receiver designers to assure proper locking and consequently proper decoding. This problem is especially acute in spread spectrum communications because of the wide bandwidth needed to achieve the desired process gain 12001.

The loss of coherence bandwidth with the increased scintillation activity has been experimentally documented by the Wideband Satellite data [ 201. A comb of seven equally spaced frequencies (413.02 5 11.47n MHz, where n = 0, 1,2, 3) were used to obtain the discrete two-frequency one-position mutual coherence functions under various scintillation levels. The

YEH AND LIU:

(a)

RADIO WAVE SCINTILLATIONS IN THE IONOSPHE :RE 3 5 5

[207] show that these sharp gradients can indeed be the sources of large amplitude fluctuations beyond those that can be predicted by using the stochastic theory. Large fluctuations in amplitude are apparently caused by scattering and diffraction near the sharp gradients and are especially apparent at gigahertz frequencies. The deterministic approach to the problem is new, and further investigations in this area are desirable.

Section I1 reviews structures of ionospheric irregularities. Exuerimental evidence for their existence is abundant. How-

r - T T - - T 7 -e - --.T -&- ever, the quantitative characterization of these irregularities

TI.iIE DELA 1’ (CHIPC)

Fig. 31. The correlator output in a direct sequency spread spectrum re-

(a) The coherence bandwidth of the ionospheric propagation channel ceiver as a function of time for three levels of scintillation activities.

is comparable to the PN spread spectrum signal bandwidth. (b) The

(c) The coherence bandwidth is about one-tenth of the signal band- coherence bandwidth is about one-fourth of the signal bandwidth.

width. (After Bognsch er al. [ 1991.)

results show good agreement with the theory [ 2011. Using these experimentally obtained discrete mutual coherence functions, the approximate waveform of a synthesized pulse is obtained, from which the mean time delay can be computed and checked against the theory. Again good agreement has been obtained.

V I . CONCLUSION We have reviewed in this paper both theories and experiments

of radio wave scintillations after transversing the ionosphere. This review attempts to update the information previously reviewed using radio star observations [38] and satellite transmissions [ 391, [ 2021, [ 2031. This review opens with a discussion of the motivation for stochastic formulation, on which the scintillation theory is based. Even though the scintillation theory agrees in a large measure with the experimental results, there are areas in which the stochastic approach may not be fruitful. Irregularities inside the equatorial bubbles show wedgelike structures with steep gradients [ 2041, [ 2051. These steep gradients show high coherence and do not appear to be statistically homogeneous. Crain er ai. [206] note that these sharp gradients can provide a refractive scattering mechanism whereby gigahertz scintillation can be explained. By numerically solving the parabolic equation in a deterministic ion* sphere model constructed from the in-situ data, W e d et ai.

is improving. For applications in which the frozen flow assumption is adequate, the three-dimensional spectrum is known to be of the form @ N ( K ) Q K - ~ with p = 4 1 [421, [ 1571. There is some evidence to indicate the dependence of the spectral index p on the strength of fluctuations or the local time in the equatorial region. The outer scale, if it exists, is not known, although tens of kilometers have been used in many computations. It is possible that other prevailing ionospheric perburbations of different origins, such as traveling ionospheric disturbances, may have prevented the true outer scale from experimental disclosure. On the other hand, radar measurements [ 541, [ 551 strongly suggest the existence of an inner scak-and a possible change of spectral law far yery small irregularities. Very liffle is known a b u t the nonfrtJzen flow nature of the ionospheric irregularities. Radar returns show a Doppler shift, indicative of the ionospheric drift, and a Doppler broadening, indicative of turbulent motion. The review of Section I1 stresses irregularity characterization, but it does not touch the production mechanisms for these irregularities. It is generally believed that some plasma instability mechanism is responsible for initiating irregularities. This subject matter is covered in several reviews [48], [208], [ 2091.

Section 111 is concerned with the scintillation theories. Ionospheric scintillations differ from the usual discussion of random medium propagation in at least two aspects. The first difference is that the background medium is highly dispersive, so that severity of scintillation is frequency dependent. The second difference is the propagation geometry. Since the ionospheric irregularities can be anisotropic, the propagation path can make an arbitrary angle relative to the irregularities. Also the existence of free space below the ionosphere allows diffraction effects to take place after the wave has traversed the irregular region. The theories under weak-scatter regime seem to be well at hand. The S4 scintillation index, the mutual coherence functions, and the spectra can be computed using the phase screen or Rytov approximations as long as the scintillation is weak. When the scintillation is strong, some of the moment equations are difficult to solve. Some numerical solutions do exist and they have been helpful in indicating the nature of behavior in the strong scatter regime. Work is still needed to develop a comprehensive theory which would indicate regorously the probability distribution of the scintillating signal. Several heuristic arguments have been used in support of some probability distribution models. Chi-square tests of these models have been devised using experimental data. The results as reviewed in Section IV indicate the preference of a bivariate distribution that would reduce to the product of a Nakagami-m distribution for the intensity and a normal distribution for the phase in the limit of zero correlation between the two variates [ 1671.

In addition to reviewing the signal statistics in Section IV, other experimental results have also been summarized. Experi-

356

mentally measured or deduced irregularity structures are summarized first. The structure information includes not only irregularity spectra but also the dimensions of irregularity patches and the propagation geometry. In the weak scintilla tion limit, a frequency law of the form S4 af-”, n = ( p + 2)/4, has been confirmed. For any level of scintillation a universal solution can be constructed by adopting the Rytov scintillation index S40 as an ordering parameter [ 11 51. In this way the true scintillation index S4 can be scaled from the universal curve if the Rytov scintillation index S40 is known.

Because of the saturation of scintillation due to multiple scattering, the spectral index n for the frequency dependence of the scintillation index becomes frequency dependent itself. Experimental results are shown to agree with theoretical predictions. Phase fluctuations seem to follow the geometric optics result which predicts the l/f behavior for rms fluctuations of the phase. The data show that except for intense scintillation cases, the departure from l/f dependence is not appreciable. This is the consequence of the dominance of phase fluctuations in ionospheric scintillations by large-scale irregularities.

The correlation distance (or coherence time) of the intensity fluctuations is controlled by two competing factors: Fresnel filtering and multiple scattering. In the weak scintillation regime, the f-’/’ dependence of the correlation distance is an indication that the controlling factor is the Fresnel filtering. As the frequency decreases, decorrelation due to multiple scattering becomes important and the correlation distance decreases with frequency. The maximum correlation distance will occur at a frequency somewhere between those two limiting cases. Experimental data clearly bear out this prediction. For phmeflwfCtUgfhns, since the Fresnel filtering is not effee tive, the correlation time is almost a constant with possible decomhtion effects at the low end of the frequency spectrum when multiple scattering becomes important.

The power spectra for the intensity, the phase as well as the quadrature components of the field are also studied in Section IV. In the case of weak scintillation, the theoretical predictions agree very well with observational results. For strong scintillations, however, only qualitative comparisons are possible.

Some experimental results on the average field and the coherence function are also presented in Section IV and compared with theoretical computations.

In studying the temporal behavior of the signal in Section V, the importance of one-position two-frequency mutual coherence function is stressed. One convenient way of describing the temporal behavior is to use temporal moments. For many applications one is interested only in quantities such as the mean arrival time, the mean pulsewidth, and the pulse skewness. These quantities can be related to temporal moments. A method developed to compute these temporal moments is reviewed in Section V. Although the method is general and capable, at least formally, of computing temporal moments to any order, the algebraic complexities increase rapidly as one moves to higher order moments. At present, analytical expressions up to the fifth order have been derived in the literature. Once the beginning few moments are known, an orthogonal polynomial expansion technique can be used to construct the approximate mean pulse shape. Very few data sets are available against which accurate comparison can be made to verify these theoretical results. Some numerical experiments are available, which suggest good agreement.

PROCEEDINGS O F THE IEEE, VOL. 70, NO. 4, APRIL 1982

The subject matter reviewed in this paper is expected to have applications in transionospheric communications. These applications are only touched upon in this review. Interested readers should consult several publications [ 199 I , [ 2 101.

Another topic that has not been included in this review is the subject of generating mechanisms for the ionospheric irregularities. There are fast developing activities in that area. Interesting readers are referred to several recent review papers on that subject 1481, [2091.

It is fair to state that, thanks to the coordinated observational programs and advances in theoretical studies, our understanding of the phenomenon of radio wave scintillation in the ionosphere has improved quite significantly. We are now in a better position to apply this knowledge to the two major areas mentioned in the Introduction that have helped to sustain the interests of ionospheric scintillation study. The first is the application to improve the transionospheric communication systems, involving channel modeling prediction, design of adaptive system, etc. The other area is the application to the investigation of the geophysical processes that generates these irregularities. This will involve the search for more effective methods to deduce from scintillation data the important geophysical parameters, in other words, the inversion problem, Future progresses is expected in these two directions,

ACKNOWLEDGMENT C. H. Liu would like to thank the Department of Electrical

Engineering, National Taiwan University, for their hospitality and assistance during his stay in the Department where part of the work was carried out.

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Global Morphology of Ionospheric Scintillations JLES AARONS, FELLOW, IEEE

Invited Paper

Abmct-Starting with post World W u I1 studies of fading of radio s t a r sources and continuing with fading of satellite signals of Sputnik, vast quantities of data have built up on the effect of ionospheric irregularities on signals from beyond the F layer. The review attempts to organize the available amplitude and phase scintillation data into equatorial, mid&, and high4atitude morphdogies The effect of magnetic activity, solar sunspot cycle, and time of day is shown for each of these three latitudinal sectors.

The effect of the very high levels of solar flux during the past sunspot maximum of 1979-1981 is stressed During these years unusually hi@ levels of scintillation were noted near the peak of the Appleton qua- torial anomaly (- +15” away from the magnetic equator) as wen as over polar latitudes. New data on phase fluctuations are summarized for the auroral zone with its sheet-like irregularity structure.

Manuscript received October 19, 1981; revised February 2, 1982. The author was with the Air Force Geophysics Laboratory, Hanscom

AFB, MA 01731. He is now with the Department of Astronomy, Boston University, Boston, MA.

One m d is now available which will yield amplitude and phase predictions for varying sites and solar conditions. Other models, more limited m their output and use, are also available. The models are out- lined with their limitations and data bases noted.

New advances m morphology and m understanding the physics of irre%uity development in the equatorial and auroral regions have taken place. Questions and unknowns in morphology and in the physics of heguhity development remain. These include the origin of the Beeding sources of equatorinl irreguluities, the physics of development of auroral irresulority patches, and the morphdogy of F-layer irregularities at middle latitudes.

A I. INTRODUCTION

RADIO WAVE traversing the upper and lower atmosphere of the earth suffers a distortion of phase and amplitude. When it traverses drifting ionospheric

irregularities, the radio wave experiences fading and phase fluctuation which vary widely with frequency, magnetic and

U. S. Government work not protected by U. S. copyright

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