Ionospheric scintillation calculations based on in situ irregularity spectra

Radio Science, Volume 12, Number 5, pages 797-809, September-October 1977

Ionospheric scintillation calculations based on in situ irregularity spectra

Etnanoel Costa and M. C. Kelley

School o• Electrical Engineering, Cornell University, Ithaca, New York 14853

(Received January 19, 1977.)

Recent results from rocket, satellite, and radar experiments have greatly increased our understanding of equatorial spread-F and its effects upon communication systems. In situ measurements have shown that the typical irregularity spectrum is a power law with a one-dimensional index of -2. Previous applications of scintillation theory to such irregularities have utilized an anisotropic power law, in an effort to model elongation of irregularities along the magnetic field, and have found approximate solutions for the scintillation index S 4, the rms phase deviation, and the characteristic scale size of the scintillation spectrum. We present here rigorous closed-form solutions for these quantities which are valid to the extent that weak scattering thin screen theory allows. In addition we introduce a second "hybrid" form for irregularity spectra which is gaussian along the magnetic field and a power law in the perpendicular plane. The index is chosen in such a way that a one-dimensional spectrum obtained on a spacecraft whose velocity •s makes a reasonable angle to the magnetic field and varies as k -2. Such a spectrum is introduced since it seems likely that at least at long wavelengths equatorial spread-F is an interchange instability and that the density variation along /• is that of the zero-order density variation. One-dimensional spectra for small angles between •s and/• would hence be steeper than k -2 at intermediate k values, a result consistent with some in situ measurements. If particle precipitation is responsible for high latitude irregularities at long wavelengths, the hybrid spectrum might also be more appropriate for their characterization. At longer k, waves with small but finite k z such as drift waves might be important and hence the anisotropic power law more appropriate. Using both spectral forms, curves are presented for S 4 and the characteristic scale as a function of x = 2 k•/k•, where k o is the outer scale wave number of the irregularity spectrum and k I the Fresnel wave number. The frequency dependence of S 4 based on the power law spectrum is also plotted in the regime where the theory is applicable.

INTRODUCTION

New insights into the physics of the equatorial F region ionosphere have accrued from a variety of recent rocket [Kelley et al., 1976a; Morse et al., 1976], radar [Balsley et al., 1972; Woodman and LaHoz, 1976], satellite [Kelley 1972; Kelley and Mozer, 1972; Dyson et al., 1974], and theoretical studies [ Haerendel, 1974; Hudson and Kennel, 1975; Chaturvedi and Kaw, 1975; Scannapieco and Ossa- kow, 1976]. Due to the increased understanding of the medium, there has been a parallel upsurge in attempts to quantitatively apply these new ideas to the phenomena of radio wave scintillation. The equatorial region has been emphasized since it displays the most intense scintillation levels which extend even into the gigahertz frequency range [Cra[t and Westerlund, 1972; Taur, 1973]. Costa and Kelley [1976] showed that upwelling regions

Copyright (•) 1977 by the American Geophysical Union.

of depleted plasma density (which seem to occur at the equator and which are described in the articles referenced above) might break up into intense irregularities with sufficient power in the important region of wave number space to cause gigahertz scintillations. Basu and Basu [1976] have shown that regions extended in altitude as much as those observed by topside satellites and by the Jicamarca radar would also be capable of producing the gigahertz and saturated VHF scintillation observed at the equator. Basu and Kelley [1977] reviewed these and other recent results and suggested a model which unifies the two ideas in an attempt to explain both the rapid onset of intense scintillation and its several hour duration in terms of the rapid upwelling of depleted plasma density regions. Si- multaneous scintillation and backscatter measure-

ments [Basu et al., 1977] have confirmed the predicted close correlation between saturated VHF and topside plumes.

The purpose of the present paper is to extend

797

798 COSTA AND KELLEY

previous work by Briggs and Phillips [ 1950], Briggs and Parkin [1963], Budden [1965], Singleton [1970], and Rufenach [1974, 1975a] in characterizing the medium and its scintillation effects based upon in situ measurements. We do not attempt to extend the discussion into the multiple scattering regime but merely to apply the weak scattering thin film screen theory in a mathematically rigorous way. We also introduce a new way of characterizing the irregularities in the medium which may be more physically meaningful in some wavelength regimes than previous models.

In the next two sections we develop two irregularity models consistent with in situ observations. We then apply the theory to obtain closed-form solutions for such quantities as the rms phase deviation, the scintillation index S 4 , and the characteristic scale of the scintillation spectrum.

IRREGULARITY DESCRIPTION

Assuming that the density fluctuation N(x) is a zero-mean stationary random process, we define the spatial autocorrelation function oN(r) of the density fluctuation by

AN 2 p•(r) = {N(x + r)N(x)) (1)

where x - (x,y,z) is the position vector, r - (r•,ry.,r•) is the correlation lag vector, AN = (N 2 (X)) •72 is the rms density fluctuation, and averaging is denoted by the angular brackets. Note, from the above definition, that p•(0) - 1.

The relationships between the three-dimensional power spectrum of the density fluctuations P•(k) and the autocorrelation function p•(r) are given by the Fourier transform pair

l fff AN 2 p•(r) exp(-ik ß r)dr (2) Ptq (k) - (2 'rr)3

AN2ptq(r)=fff Ptq(k)exp(ik,r)dk (3) where k = (kx,ky,k z) is the spatial wave number vector.

Original studies of scintillation employed gaussian irregularities with some dominant scale size. However, far and away the most commonly observed one-dimensional spectra of density irregu-

larities in the ionosphere display a power law varying as k -2 over a wide range of k values. This spectrum has been observed in the high latitude F region [Dysen et al., 1974; Phelps and Sagalyn, 1974] as well as the equatorial F region [Dyson et al., 1974; Kelley et al., 1976a; Morse et al., 1976]. Most physical processes one can think of in a low [3 plasma such as the Earth's ionosphere are two dimensional and one is therefore led to allow dif-

ferent variations parallel and perpendicular to B. Furthermore, results of many ground-based experiments have shown that F-region irregularities are field aligned. Early results from radio star scintillations showed that the axial ratio was 7 to 1 or

greater for 0.2 -< ks -< 0.85 km [Kester, 1963]. Improved methods have shown elongation of the order of 60 to 1 in the same interval of perpendicular wavelengths [Kester, 1966]. Radar measurements set a lower limit of 10 2 on the axial ratio at h x = 3 m [Hudson and Kennel, 1975].

Thus it was natural to introduce an anisotropic power law spectrum for ionospheric irregularities [Ru[enach, 1975a] which is one of the two forms we discuss in this paper and which we define as follows'

P•v(k') = [ot•vsin(3,tr/•v)/4,tr 2 k•]AN •

ß {1 + [(k'•: + k;:' + ot:k':)/k•]•'/2} -• (4)

in a reference frame with k' z axis aligned with the magnetic field direction, as shown in Figure 1 (which also defines all the reference frames we will use

later). In this expression, •/v is the spectral index, e• is the axial ratio and k o is the intrinsic outer scale wave number. The anisotropic power law spectrum is essentially flat for Ikl < ko and it falls off approximately as Ikl for [kl > /Co. This spectral form is essentially the same assumed by Rufenach [1975a] except that we kept a general spectral index at this stage.

There are several reasons to introduce a second

type of irregularity description, which does not involve a power law dependence along the magnetic field. Consider first the high latitude irregularities. Unlike the equatorial case it has proven very diffi- cult to postulate an F-region plasma instability which operates over the wide range of geophysical conditions in which irregularities are found. It has been suggested [Piddingten, 1964; Kelley et al., 1976b] that the irregularities are produced in the auroral oval and are convected throughout the

WEST

INCIDENT WAVE

Bo

UP z,kz / i i

k z ,z

i

kx,kx,x,x

ky,y

! •ky,y

,SOUTH

Fig. 1. Reference frames.

high-latitude regions by the dc electric field driven plasma drift. In this model individual flux tubes filled with plasma due to ionization in the auroral oval behave independently due to the very slow diffusion across the magnetic field. The plasma distribution along the magnetic field would thus be more like a Chapman distribution, at least at large perpendicular wavelengths. Likewise at the equator the longest wavelength waves may be due to a flute mode which has kll zero [Dungey, 1956; Hudson and Kennel, 1975]. Variations in density along the field line then would be more characteristic of the zero-order density profile along that field line before the interchange took place.

We thus introduce a second spectral shape which, defined in the same reference frame is

PN.(k') = [('y. - 2)a/2'rr 3/2 k•] AN • { 1 + [(k'x •

+ k'•2)/k•)]} -'•./2 exp [-(ako)2(k'•/ko): ] ($) This "hybrid" power spectrum is semi-separable, that is, PN•(k') = F(k' z) G(k'•,,ky), with a gaussian distribution along the magnetic field direction and a power law variation in the perpendicular plane. The parameters •/• and k o have the same interpretation as before while the role of a -•, along the direction of the magnetic field, is similar to that of k o in the perpendicular plane.

The normalization factors inside square brackets

IONOSPHERIC SCINTILLATION CALCULATIONS 799

in (4) and (5) are such that

AN2:•••PN(k')dk ' (6)

This relation adjusts the spectral forms in such a way that they become consistent with (3) evaluated atr = 0.

Both spectra describe irregularities aligned with the magnetic field and with a circular cross section in the transverse plane. In the next section we show that by proper adjustment of the spectral index both spectral forms can yield one-dimensional power spectra which are consistent with in situ measurements. In order to improve our understanding of the properties of the assumed spectra, and anticipating the results presented below somewhat, curves of constant normalized spectral density (isospectroids) in the plane k• 0 are shown in Figure 2 for •/p = 4 and •/• = 3. These isospectroids have been obtained from (4) and (5), and are defined by

ISOSPECTROIDS

k;:/ko 8

ß

k'x/k o I O• ø - k•/k o C=0.5

Fig. 2. Isospectroids based on the power law (thin lines) and "hybrid" (thick lines) spectra, for two values of spectral density (C = 0.5 and C = 0.005) and for different values of a and ak o. Only half of the total isospectroid is plotted, since the total isospectroid is symmetric with respect to the axis, k' z/k o.


L,O,'•,O,k'•) = {1 + [•k'•/ko? + o,'•k'•/ko?]'} -'= c

(7)

[u(k'x,O,k' •) = [1 + (k'x/ko)2] -3/2

ß exp [-(ako? (k• / ko): ] = X/-•-C/X/-•-• (8) The parameter C determines the level of spectral density corresponding to each spectroid and the curves are normalized in such a way that, for each value of C in Figure 2, the spectral densities of the two spectra are equal Pm,(k'x,O,k•) = PN•(k'x,O,k' z) when a = (ako). Thus, not only the changes in the isospectroids (obtained from the same spectrum) with a [or (ako) ] and C can be observed, but power law and "hybrid" isospectroids can be compared when a = (ako). We note that the shapes of the power law and the "hybrid" isospectroids are remarkably similar for C = 0.5 and a = (ako), and that they differ substantially for C = 0.005 and •x = (ako). Although not obvious, for infinitely elongated irregularities (a,a--• c•) and for •/t, = 4 and •/• = 3, the spectra (4) and (5) become asymptotically equivalent. A proof of this statement is sketched in appendix A.

ONE-DIMENSIONAL SPECTRA OBTAINED FROM IN

SITU OBSERVATIONS

As shown by Fredricks and Coroniti [ 1976], in situ measurements of density irregularities cannot, in general, be analyzed in terms of wave number power spectra. However, satellite and rocket probes fly across the irregularity region with extremely large velocities. Thus, it is possible, to a good approximation, to assume that the density fluctuations are entirely spatial. Fredricks and Coroniti show that under this assumption one can sensibly Fourier analyze the time series obtained from the probe measurements into a one-dimensional power spectrum as a function of the frequency in the spacecraft frame and then to deduce a one-dimensional wave number power spectrum by the relation k = to/V s , where V s is the probe velocity [Lovelace et al., 1970; Cronyn, 1970]. The discussion below is thus restricted to this approximation.

Assume, i•itially, that a probe is flying perpen- dicularly to the magnetic field direction. This is approximately the situation we have when a rocket is flying vertically in the equatorial region or a satellite moving horizontally at high latitudes. Even at the equator a polar orbiting satellite usually moves

across the magnetic field at a sizeable velocity. Since both spectra P•t,(k') and P•n(k') are symme-

' and k' tric in k• y, we can assume that the probe is flying in the y' direction, without loss of generali- ty. Then the probe will only sample spatial Fourier components along the y' axis, and the spatial spectrum it sees will be the integral of P•(k')over the components transverse to the flight direction (k'• and k•, in our case). That is, the one dimensional spectrum seen by the probe is

k' k•) k•dk (9)

Substituting (4) into (9) we obtain the one dimensional spectrum seen by a probe, based on anisotropic power law irregularities

sin (3 •r/-V •,) AN2 ' = 'k'/ko) (10) PNp(k y) (1 / 2) • P(•I p, y

sin (2•r/%,) k o where

p(•!;x)=l_sin(2xt/•!)•12 dy (2•r/•) 1 + y•/2

_ sin(2•r/•/) •+• dy - (2•r/•/) •2 1 + y.•/2 (11) Following the same procedure for the hybrid irregularity form given by expression (5), we obtain

!

PN•(k y) = (•/u- 2) r[(•/u- 1)/2] AN:

2 X/'--• r(•u/2) k o !

ß [1 + (ky/ko)2]-('•. -ø/2 (12)

where F (z) is the gamma function. We note here an interesting result, which has

also been discussed by Cronyn [1970], and Rufen- ach [1972]. Using the last part of expression (11), itis easily shown that P(%,;k'y / k o) is asymptotically proportional to (k'y / ko) -(•P-2) as (k'y / k o) becomes large. That is, the spectral index of the one-dimensional power spectrum PNt,(ky) is two units smaller than the spectral index of Pm,(k'). Similarly, expression (12) shows that P•n(ky) for ky >> k o. For the 'hybrid spectrum, the difference between the two spectral indices is equal to one. Thus, the one-dimensional spectrum becomes flatter than the respective three-dimensional spectrum, in both cases. Note that the difference

between the forms of the three-dimensional and

one-dimensional spectra depends on the particular choice of the three-dimensional spectral shape. A gaussian shape for example, remains unchanged in this process [Ru[enach, 1975a].

The spectral indices of the one-dimensional power spectra (10) and (12) must be adjusted in such a way that these expressions exhibit asymptotic behaviors at large k which are consistent with the results of in situ measurements. Results of rocket

and satellite observations of density irregularities in the equatorial Fregion [Dyson et al., 1974; Phelps and Sagalyn, 1974; Kelley et al., 1976a; Morse et al., 1976] have shown one-dimensional power spectra which fall off approximately as k -2 for large values of k. In order to model these results, and to predict the behavior of the rms phase fluctuation and of the scintillation index based on such ob-

servations, we must take % = 4 in all expressions related with the power law irregularities. In this case, (10) becomes

1 AN• [ 2 )•] 1 - • arc tan (k'• / k o P•(k'•)- 2%/_ •_ ko 1 AN •

2V•- k o 1 AN 2

2X/-•- k o (13)

1- -- (k'y/ ko) 2 , k'y << k o

• (k'• / ko) -:, k' >> k 0 y

Similarly, (12) shows that the appropriate value for this spectral index of the three-dimensional "hybrid" spectrum is •/a -- 3. In this case, we find

P•.(k•) = (1/•)(AN2/ko)[1 + (k'y/ko)2] -1 (14)

Therefore, (13) and (14) show that %, = 4 and •/a = 3 provides us with theoretical three-dimensional spectra which are consistent with the results of most of the existing observations. Whenever possible, however, we will present general expressions in the following sections.

Finally, it should be noted that the "hybrid" spectrum predicts that a probe flying exactly along the magnetic field lines direction would detect a one-dimensional gaussian spectral shape, while the power law spectrum would still yield a k -2 one- dimensional spectral shape. If the probe is flying at any other angle with respect to the magnetic field direction, the expressions for the one-dimen-


sional spectra become more complicated. The one dimensional power law spectrum would still be given by a k-'• variation, but it can be shown that the one-dimensional spectrum derived from the three- dimensional "hybrid" spectrum is given by

Pt•u(k;) = (a AN2/•cos 0) Im {Z[(ak o)

ß secO(k'y/ko) + i(ako)tanO]} (15)

where Z(s) is the plasma dispersion function of the complex argument s, as defined by Fried and Conte[ 1961], and 0 is the angle between the flight direction and the direction of the magnetic field. This expression reduces to expression (14) when 0 = •r/2 and to

P•H (k • ) = ( a AN: / •) e -(ak ;): (16) when 0 = 0.

The plasma dispersion function is approximately given by

i• e-S:, for Is[ << 1 Z(s) • (17) -s, for Isl >> 1

Taking the imaginary parts of the above approximations and substituting the results for Im[Z(s)] into expression (15) we find that PNa(k•) tends either to a gaussian shape or to a k -2 power law depending on the magnitude of the parameter

l• = (ako)[SeceO(k ' /ko) • + tane0] 1/e y

= (ako){[(ky/ko) 2 + 1] sec20 - 1}l/2

If ½ << 1, the spectrum takes a gaussian form and when • >> 1, the k -• power law takes over. We note that ½ depends not only on the angle 0, but also on the parameter (ako) and on the range of wave numbers k y in which the measurements have been made. Even for small nonzero angles 0 (sec 0 --• 1) and moderate values of (ako)[ak o --• 1], the parameter ½ can be large. That is, based on a three-dimensional "hybrid" spectrum, we predict that a probe flying along a direction almost parallel to the magnetic field would detect a k -2 power law spectral shape for wave numbers k y much larger than the outer scale wave number k o. In the upper curve of Figure 3 we have plotted PN•(ky) versus (ky/ko) for 0 = 4 ø and (ako) = 1.43 in order to illustrate better the nature of expression (15) in the small 0 regime. In this particular example, the figure shows that P•u(k'y) is approximately propor-


•>,10 -i --

I- ß oo-e \ \ - ..o / --Hybrid t-D spectrum \ \

: _

lt• 2 i0 -I i0 ø I0 • ky/k o

Fig. 3. Normalized one-dimensional "hybrid" spectrum for 0 = cos-I(O ß •) = 4 ø and ak o = 1.43. The OGO-6 data have been reproduced from Figure 6(a) of Dyson et al. [ 1974]. The

curves have been superposed by "inspection."

!

tional to k -2'ø7 in the intermediate region 3 < (ky / k o) < 10. For smaller k the spectrum is considerably steeper than a power law would have yielded. The data points are reproduced from Figure 6 of Dyson et al. [ 1974] and correspond to "sinusoidal" spectra sometimes observed near the magnetic equator. The data points fit the upper curve very well in the

!

range k o < k y < 8ko, but approach the peak from the right more steeply than does the curve. (Note that no attempt has been made in the mathematical representation to model a turnover of any spectra at small k.)

Basu and Basu [1976] have noticed that the "sinusoidal" OGO-6 equatorial spectra seem to occur when the orbit trajectory most nearly lies in a magnetic meridian plane and hence perhaps that they correspond to minimum angles between the velocity vector and the magnetic field direction. Although not definitive, the present results show that allowing a different three-dimensional spectrum parallel to B can produce distorted (relative to power law) one-dimensional spectra at small angles. The observed spectra are more monochromatic than our hybrid spectrum, which suggests that some process might pick out a preferred parallel wave number. The OGO-6 data can also be interpreted as a nearly

monochromatic peak superimposed on a broader background spectrum. The lower curve in Figure 3 fits the theoretical spectrum to the background spectrum. The fit is still quite good in form but the theoretical spectrum in this case falls off at

!

too small a value of ky/k o. As discussed above, a non-power-law spectrum

parallel to B is attractive due to the evidence for interchange modes at small k. There is theoretical [Chaturvedi and Kaw, 1976] and experimental [Kelley et al., 1977] evidence that drift mode waves determine the spectrum at large k. Since such waves have small but finite kll, an anisotropic power law at large k might be more appropriate. A "super-hybrid" spectrum could be the answer for a break in the three-dimensional spectral index at some intermediate k value and simultaneous change from gaussian to power law along /•. Two-dimensional simulation of plasma turbulence in a magnetic field has shown such a break in the spectral form [Seyler et al. , 1975].

RMS PHASE FLUCTUATIONS, SCINTILLATION INDICES, AND CHARACTERISTIC SCALES BASED ON

THE ASSUMED SPECTRA

The intensity scintillation of radio waves passing through the ionosphere has been studied by several authors through the weak scattering, thin phase screen theory approximation: the region of the ionosphere which causes the fluctuations (assumed to have a width L) is replaced by a plane surface along which the phase of the wave fluctuates but the amplitude remains constant. If L is sufficiently small, most of the intensity fluctuations are built up from the phase fluctuations after traversal through the irregularity region [Salpeter, 1967].

Let P, (kx,ky) be the power spectrum of the phase fluctuations in the wave emerging from the irregularity layer and Pt(kx,ky) be the power spectrum of the intensity fluctuations in the received signal, expressed in a reference frame with k z axis aligned with the direction of the incident wave (see Figure 1). It has been shown [Salpeter, 1967; Lovelace et al., 1970; RuJ'enach, 1975a] that the following relations apply in the weak scattering limit

P,(kx,ky) = 2-rr(r e X)2 (Lsec 0) PN(kx,ky,k z = O)

P•(kx,ky) = 4sin 2 [(k• + k:)/k•] P, (kx,ky k = 0) y • Z

(18)

(19)


where k 2= 4,r/hz, h is the wavelength of the incident wave, z is the distance (measured along the ray path) from the source to the observation point, 0 is the angle between the ray path and the vertical at the point where the ray path intercepts the phase screen and r e is the classical electron radius. Equations (18) and (19), and thus all the results in the remainder of this paper, assume h << a << L << h and q)o << 1, where a is the correlation length of the phase fluctuations, q)o is the rms phase fluctuations, defined in the next paragraph, and h is the height of the irregularity layer [Salpeter, 1967].

We define the rms phase fluctuation q)o and the scintillation index S 4, which is the normalized rms intensity fluctuation, by

q)o 2 = P, (k x ,k•)dk x dk

$• = Pt(k x,k•)dk• dk•

(20)

(21)

The power spectra of the density fluctuations (4) and (5) are defined in a reference frame with the k'• axis oriented in the magnetic field direction, as shown in Figure 1. They can be expressed in the unprimed reference frame (k• axis along the incident wave direction) through the rotation

ky ky cos q• - k• sinq• (22)

k' z = kysinO + kzcos •

where q• is the angle between the magnetic field and the ray path. Substituting the transformation (22) into expressions (4) and (5), we obtain

PNp(k x ,ky ,kz) = ot 'yv sin (3'rr/-yv) AN 2

4,11 '2 ko 3

. [1 + (k2x+ [•2yk2 +[•2k2 +or kyk y z z yz z 'yp/2 (23)

where 132y = cos2• + a2sin2q•, 13• = sin2• + at 2 COS 2 t•, et yz = (at 2 _ 1)sin(2 •), and

PNt-t(k,,,ky,kz) = [('Yt-t- 2)/2'rr3/2 ] a(AN2/ko 2)

ß exp{-(ako) 2 [sin2q• k 2 + cos2t• k 2 + sin(2•)kyk ] y z z

+ ko2}/{1 + [k• 2 + cos2•k 2 + sin2q• k 2 y z

- sin(2•)kykz]/ko2}•,•/2 (24)

The rms phase fluctuation q)oebased on the power law spectrum is obtained from (20) and (23)'

sin (3 xr/h,p) ot AN 2 q)o2p = xr (rex)2 (L sec0) • (25)

sin(2xr/•/p) [3 y k o

Similary, the rms phase fluctuation q)o• based on the "hybrid" spectrum is obtained from (20) and (24):

2• F[('y H - 1)/2] [cos q,[ F[(•IH- 2)/2]

aU(l / 2,2 - •t n/2, a2 k2o tan2 •)(re h)2 (L sec 0) AN 2

(26)

where U(a,b,x) is a confluent hypergeometric function [Abramowitz and Stegun, 1965]. In the equatorial F region when viewing satellites in the equatorial plane, it is a good approximation to assume • = ,r/2. The rms phase fluctuation based on the "hybrid" spectrum then becomes

q)o2•(q• xr/2) X/•' (•/• 2)F (•/•- I )/ = = - r (•,•/2) 2

ß (re X )2 (L sec 0)AN 2/k o (27)

Except for approximations which are inherent to the weak scattering theory, expressions (25) and (26) are completely general. In both cases, the rms phase fluctuation q)o is directly proportional to the rms density fluctuations AN and to the square root of the distance the wave travels inside the irregularity region (L sec 0); and it is inversely proportional to the wave frequency 1' = c/X and to the square root of the intrinsic outer-scale wave number k o (for the "hybrid" spectrum, this is true when • = ,r/2). Finally when %, = 4, • = 3, and • = ,r/2, expressions (25) and (27) show that (!)o• is greater than q)oe by only 11%.

In order to derive the expressions for the scintillation index S4p based on the power law irregularities, we assume that %, = 4 in expression (23). In this case, the power law spectrum (4) reduces to the expression assumed by RuJ'enach [1975a]. Carrying out the required integrations with the help of the table by Gradshteyn and Ryzhik [1965], expressions (18), (19), (21), and (23) give


S4p -- V•'l•opFp([• y;4;x) where

(28)

F•,(13;4;x) = l - Io [( f32 -1)x] 2•3 2

- x (29) ß exp 2132 x = 2k•/k•, and Io[y] is the modified Bessel function of first kind and zero order. Expression (28) reduces to the already known result for the isotropic case [Ruyenach, 1975a] when 13y = a = 1. However, unlike previous treatments we did not have to make any approximation in order to treat the anisotropic case. The function Fp (13 ;4;x) is plotted in Figure 4, which shows that Fp( 13 ;4;x) is independent of 13 for large values of this parameter. We see that, as predicted by Ruyenach [ 1975a ], all curves have the same slope for small values of x and that they tend to one for large values of x. However, the curves of Figure 4 are closer together than those predicted by Ru[enach [ 1975a ], indicating that the actual dependence of S 4 on 13 y is not fully described by Rufenach's "gaussian geometric factor" approximation.

In discussing the expression for the scintillation index S4i. I based on the "hybrid" irregularities, we are going to assume q• = ,r/2, and hence primarily consider the equatorial ionospheric region. Unfor- tunately, it has not been possible to derive an expression for S4• assuming a general angle q•. From expressions (18), (19), (21), and (24), we obtain

S4i_ I -- X/-•q> oH FH (ak o ;•/H ;x) (30) where

[ [ [ [ ] [ , • [ , [ ] [ [ [ [

¾

io I

i i , , I , i , , I , i i i I i(•z iO-• •o o •o •

x = 2 (ko/kf) z

Fig. 4. Normalized power law scintillation index S4p/'•q)Op versus x for different values of 15.

Fig. 5. Normalized "hybrid" scintillation index S4H/'•V/•(•OH versus x for different values of ak o.

F•(ako;•lH;x) = 1-r(•./2)/r[(•.- 1)/2]

Re[e -'ø, U(1/2,(3 -•t•)/2, ix)]

[1 q- X2/(ako)4] 1/4 (31)

x = 2k•/k•, 0, = (1/2) arctan[x/(ako)2], and the real part of the expression inside the square brackets in (31) must be taken. This expression does not constrain the spectral index •H to any particular value and could be used to study the dependence of S4• on the spectral index. However, we will restrict ourselves to • = 3. In this case, expression (31) becomes

F•(ako;3;x)= 1 + (,r/4)x{cos(xrx/2-O•)

ß [Jo(x/2) + Y•(x/2)] - sin(xrx/2-0•)

ß [J, (x/2)- Yo(X/2)]}/[1 + x2/(ako)4] '/2 (32)

where x and 0• have been defined above and Jo.• (Y) and Yo.• (Y) are Bessel functions of first and second kind, respectively. The function F•(ako;3;x) is plotted in Figure 5. This function is independent of (ak o) for large values of this parameter and tends to one for large values of x.

A comparison between Figures 4 and 5 indicates that the slopes of Fp(•;4;x)and F•(ako ;3 ;x) coincide for small values of x. Further, we note that F•(ako;3;x) and Fp(13;4;x) are of the same order of magnitude and that F•(ako;3;x) is generally greater than or equal to Fp(13;4;x) when both (ak o) and 13 are large.

Based on a previous comparison between the power law and the gaussian spectra [Ruyenach, 1975a ] we might have expected the "hybrid" spectral shape to yield smaller values of scintillation indices than those derived from calculations based

on a power law spectrum. However, we have shown


that when both three-dimension power spectra are consistent with the observations, they yield essentially the same value of S 4. This happens because the "hybrid" spectral density decrease along any direction perpendicular to the magnetic field lines is slower than that of the power law spectral density, due to the smaller spectral index (•tv = 4 and = 3). This fact makes up for the fast decrease, relative to a power law, in the "hybrid" spectral density along the magnetic field direction, due to the gaussian factor.

The dependence of the scintillation index S4 on the frequency • of the incident wave, which is important in the design of communication systems, can be obtained as follows. If x >> 1, expressions (28) to (32) and Figures 4 and 5 show that S4 •-(•00C f--I for both spectral forms. If x << 1, we can study the S 4 frequency dependence by substituting approximate expressions for the exponential and Bessel functions into expression (29) and (32). The final results are summarized in Table 1, where we have defined an index n(f) by S 4 -- A[ (A is independent of [). Note that n(f) depends on the frequency of the incident wave.

Assuming that n(f) is sufficiently smooth in the transition region (1 < n < 1.5), we can make the following approximation [ Yeh and Liu, 1975]'

S4 (fl) / S4 (f2) -- f] -n([l) / f• .([2) • (fl / f2 )- "([1) (33)

for • • •. Substituting expressions (28) and (30) for S 4 into expression (33) and solving for n(•), we see that nv (f) and n• (f) have the general form

In [ F 2 (f ,) / F 2 (f2) ] n (f•) = I - (1 / 2) (34)

ln(f•/f2)

where F 2 (f) stands for F•, (13 ;4;x) or F•(ak o ;3 ;x). Although it is not explicit in expression (34), nv(f) also depends on I• and k o, and n,(j• depends on (ak o) and k o, in general. Expression (34) allows us to investigate the approximate behavior of n(f) in the intermediate region. As an example, we have plotted nv(f) in Figure 6 for different values of

TABLE 1. Asymptotic frequency dependence of the indices ti.

x<< 1

Indices x >> 1 x >> [5 2 [(ako) 2 ] x << [5 2 [(ako) 2] tip f-I f--I f--3/2 till f--I f--I f--3/2

! i ! i I i i i ! i i

1.5 • i a

1.4] /• d• •.3 -

i '

WAVE FREQUENCY (MHz)

--

Fig. 6. Power law index n v versus the wave frequency [ when the frequency variation of the scintillation index S4v is represent-

ed by S4v = A vf-"vcø with A v independent of [.

13 and k o, assuming tl, = •r/2 and z = 400 km. For fixed [3, we see that n v decreases with k o. We also note that nvincreases with moderate values of 13, for the same value of k o . However, nvbecomes independent of 13 as this anisotropy factor increases.

At this point, it should be remembered that the weak scattering, thin phase screen approximation is only valid for S4 <• 0.4 [Ru[enach, 1975b ], which is an easily violated condition for typical equatorial ionospheric parameters and for frequencies less than 100 MHz (in some situations, even for frequencies in the UHF range). Thus, the curves in Figure 6 suggest that only occasionally, if ever, will the scintillation index be inversely proportional to the first power of the frequency of the incident wave in the region of validity of this theory. That is, based on the assumed spectral shapes, we expect the scintillation index dependence on the frequency of the incident wave to be fairly well represented by the law S4 o: [-3/2 in the region of validity of the weak scattering theory, which approximately corresponds to high frequencies (usually in the gigahertz frequency range) for typical parameters of the equatorial F region ionosphere.

Attempts have been made to relate the fluctuation rate of a received signal to the velocity of the medium by use of a "characteristic scintillation


scale size" at the frequency of interest. For gaussian spectra this is quite straightforward since there is a simple mapping possible between in situ and ground level scale. Using a generalization of this technique, as outlined by Rufenach [1975a] we define the first absolute moment of the one-dimen-

sional scintillation spectrum in the k• and k y direc- tions by

Q,, = Ikl P•(ku)dku, u = x,y (35)

Then we define the characteristic scintillation wave

numbers Kx and Ky by

Ku = Q•/S•, u = x,y (36)

and the characteristic scintillation scale sizes a x and a y by

a•= 2xr/K u, u=x,y (37)

The evaluation of the one-dimensional scintilla-

tion power spectrum based on the assumed density spectra (4) and (5) is very complicated. However, this calculation can be avoided if we remember the

fact that

Pt(k•) = Pt(kx,ky)dkv, u = x,y and v = x,y,

with u • v (38)

Then, we can write expression (35) in the form

Q•, = Ik•l P•(kx,k,)dk•dky, u = x,y (39)

which is more tractable.

The characteristic scintillation scale sizes a x and a y have been evaluated for both spectral shapes. From expressions (18), (19), (23), (36), (37), and (39) we find the expressions for the characteristic scale sizes based on the power law spectral shape

a•,• = ( X/-•-xr/ 4)ro F•(13;4 ;x)/ F • (l•;4;x) and

(40)

aey = (V•-'rr/4) 13 roF•,(13;4;x)/Fey(13;4;x) (41) where r o = 2w/k o, [•2 = cos2q• + a•_sin•q•, F•([l;4;x) is defined in expression (24),

F•(13;4;x) = 1 - [eW2G,(xl w) + e-W2G2(•lw)]/2xlw

+ 13 {(2xl 113)[G• (w) + G•_ (w)] + 2[arcsin(xl/13)

- (xl/1•2)] • +•rX/-•(sinhx/x- 1) +4w

- 2[G,(w) + G2(w)] + C•(13;x)}/(2X/--• xl) and

(42)

Fey(13;4;x) = 1 - [e-•G•(xlw) + eXG•_(•lw)]/(2xlw)

+ • {(2.•)[G.(x/-•) + G•(x/-•)] + 211•-' ln(•l + 1•) - •l]w + 2x{(X/-•/2)

ß [2F2(1,2;1.5,3;-x) - 2F2(1,2;1.5,3;x)] + (w/2•3 3)

ß [2F2(1,2;1.5,3;-w 2) - 2F2(1,2;1.5,3;w2)]

- Cy(13;x)}}/(2X/--•xl) (43) In expressions (42) and (43), we defined -- 1) 1/2 W = (X/•2) 1/2

z G• (z) = e z• e -'• dt (44) o

G• (z) = e -• e t• dt (45) o

:F•_(a,b;c,d;z) is a generalized hypergeometric function [Gradshteyn and Ryzhik, 1965],

cx(•.x) = 2x [• - (• - v)•/•][o.(x/ xv ) o

- Ge(• xv )]dv (46) and

c,(•.x) = [x/-T- (v - • -:)'/:] [G• (x/xv )

- G2(V' xv )]dv (47)

It should be noted that G• (z) can be expressed in terms of the error function [Abramowitz and $tegun, 1965] and that Gz (z) is proportional to the real part of the plasma dispersion function of a real argument [Fried and Conte, 1961 ]. Thus, they can be very easily computed from existing polyno- mial approximations. Also, note that C•([•,x) and Cy([5,x) tend rapidly to zero as [5 increases and can be neglected for large values of I•.

Similarly, we obtain the expressions for the characteristic scale sizes based on the "hybrid" spectral shape from expressions (18), (19), (24), (36), (37), and (39). Again, it has not been possible to obtain expressions for general angles and we re-


stricted ourselves to the case q• = •r/2.

am= ro F2 (ako;3'x)/Fm(ako;3;x) (48) H '

asy= 2•r3/2aF•(ako;3;x)/Fsy(ako;3;x) (49) where r o = 2•r/k o, F•(ako;3;x) is defined in expression (32),

Fm(ako;3;x) = 1 - [(•; + 1)/2•;2]•/2l[1- (•rx/2) •/•

ß {[1 - 2S(X/-•)] cosx - [1 - 2C(X/-•-)] sinx}]]

+ [•- •)/•)]•/• + •x/•) •/• {[• -

ß cosx + [1 - 2S(X/-•)] sinx•]] (50)

Fny(ako;3;x) = 1 - (x/4•2){[Jl (X/2) - Yo (x/2)] ß sin(-trx/2) - [Jo(x/2) + Y,(x/2)]cos(•r x/2)}

+ (x/2ako[) • {[J•(x/2) - Yo(x/2)]cos(•r x/2)

+ [Jo(x/2) + Y•(x/2)]sin(•x/2)) (51)

In expressions (50) and (51), we defined [ = [1 + x:/(ako)4] •/•, Jo,•(z) and Yo,•(z) are Bessel functions and S(z) and C(z) the Fresnel sine and cosine integrals [Gradshteyn and Ryzhik, 1965].

The asymptotic behavior of the characteristic scale sizes can be obtained by substituting approximate expressions for the special functions in the appropriate region (x << 1 or x • 1). The final results are summarized in Table 2. The asymptotic frequency behavior of the characteristic scale sizes is easily obtained from Table 2, since x For example, we find that avx, avy o: [-' for x << 1, which agrees with the approximation made by Rufenach [ 1975a ]. The characteristic scale sizes are inversely proportional to the outer scale wave number of the density fluctuations when x >> 1. However, Table 2 shows that a px and a vy (the same holds for a Hx) become independent of r o (= 2 •r / k o) when x << 1. This result indicates that it is not

possible to estimate the outer scale wave number k o of the density fluctuations from ground observa-

TABLE 2. Asymptotic dependence of the characteristic scale sizes on x.

Scale sizes x << 1 x >> 1

a • / r o x 1/2 x ø a • /13r o x •/2 x o a m / ro x •/• x ø ally / a x! x ø

CHARACTERISTIC SCALES FOR POWER LAW SPECTRUM

_ I ! I I I I t I•

• I I I I I t i i I I I I I i0 o I01 10 2

x = 2 (ko/kff

Fig. 7. Normalized power law characteristic scale sizes versus x for isotropic (13 = 1) and highly elongated (13 --) o•) irregularities. The curves for (4/X/•rl3ro)avy coincide with the curve labeled 13 = I for both the isotropic and elongated irregularity.

2/k]) << 1. Rufenach [1975a] tions when x(= 2k o predicted a linear relationship between apx. y and r o in the small-x limit, in disagreement with the above result. The present result seems physically reasonable since for small x, the characteristic scale is determined solely by the power law portion of the spectrum far above k o.

The curves for the characteristic scale sizes based

on the power law spectrum are plotted in Figure 7 for 13 - 1 or 13 ---) oo. The curves for the characteristic scale sizes based on the "hybrid" spectrum are plotted in Figure 8 for different values of (ako). There is a striking difference between

CHARACTERISTIC SCALES FOR HYBRID SPECTRUM

l' ' ' ' '1 .... I ' ' ' '1 ' ' ' 'l '

0 •

.•o

ø• ,,s'•ø l , , , ,I , , , ,I , , , ,l , • , ,I

i½ • i½ • io ø io [ lo • x:

i ! i

io :3

[6-• •o -2 ,½' [o ø [o' ,0 2 •o •

x = 2 (ko/kf) z

Fig. 8. Normalized "hybrid" characteristic scale sizes versus x for different values of ak o.


these curves and those derived from a gaussian spectral shape [Ru[enach, 1975a]' while the characteristic scale sizes based on a gaussian spectrum are constant for small values of x, the curves shown in Figures 7 and 8 decrease continuously and tend to zero as x decreases. As shown in Figure 7, the curve of normalized a ey for large values of [3 coincides with the curves based on the isotropic power law spectrum. This seems to indicate that a•/[3 is essentially independent of the anisotropy factor, and hence that a• is proportional to [3. Note, however, that a• becomes independent of [3 when the anisotropy factor increases. Figure 8 (see also Table 2) shows that a m and any display different behaviors for small values of x' the char-

acteristic scale along the magnetic field direction (any) decreases faster with x than does the transverse characteristic scale am. This difference is certainly due to the different properties of the "hybrid" spectrum along each direction. Finally, we observe that a ax = 3 apx for large values of [3 and (ak o) and for small values of x.

These exact calculations should replace the approximate values for characteristic scale size presented earlier by Ru[enach [1975a]. Since there is a dependence of scale size upon x and hence upon the distance from the receiver to the scattering region one must be careful in interpreting time variations in received signal in terms of the velocity in the medium if the height of the scattering layer is not known. Also, since there is evidence that the equatorial instability region can be highly turbu- lent in velocity as well as extended in height, use of a characteristic scale size to determine the plasma velocity is not always possible.

APPENDIX A

As the "hybrid" parameter a and the power law parameter •x tend to infinity we expect the two expressions (4) and (5) to be equivalent since in both cases the modeled irregularities are infinitely elongated along the magnetic field lines. Although not obvious at a glance, the following discussion shows that this is indeed the case.

The "hybrid" spectrum defined in expression (5) (with •/a = 3) can be written as

AN 2

PNH(k') = [1 + (k•_ /ko)2] -3/2 g(a;kll) (A1) 2•rko 2

where

k• =k '2+k '2 kll=k' x y • Z

g(a;kll) = (a/X/-•-•)e-a2k• (A2) As a • m, we note that g(a;kll) tends to zero if k • 0 and to • i• kll = 0. Further, the integral of this function over all kll is equal to 1. Therefore, g(a;kll) tends to 8(kll) as a • •, and

PN.(k')--, (AN2/2•rk•)[1 + (k•_/ko)2] -3/2 •J(kll ) as a • • (A3)

Similarly, the power law spectrum defined in expression (4) (with •/v = 4) can be written in the form

PNp(k') = (X/• / 2)(AN 2 / •r 2 ko 2) A (q) [(d ;q ;kll) where q = (k •_ / ko) 2, d = a/k o ,

J•(a;q;kll ) = {a/[1 + (q + d 2 kl•)2]}/A(q) A(q) = (•rX/-•/2)[(q 2 + 1) 3/2 + q(q2 + 1)]-1/2

(A4)

(A5)

(A6)

Initially, we note that d --, o• as a --, o•. Thus, as a • o•, we find that f(d;q;kll) tends to zero When kll :• 0 and to infinity when kll - 0. Further, expressions 2.161.1 of Gradshteyn and Ryzhik [1965] can be used to show that the integral of f(d ;q ;kll ) over all values of kll is equal to one. That is, this function also tends to 8(kll) as a --• Hence

V•' AN 2 PNt,(k') --• A (q) • (kll) as a---, oo (A7)

2 •r2ko •

It can be shown that expressions (A3) and (A7) are essentially equivalent in the region k I >> k o.

Acknowledgments. This work was sponsored under NSF grant ATM75-04269 and ONR contract NOOO15-75-C-0780. One of the authors (EC) was supported under a fellowship granted by CAPES, an office of the Brazilian government, during this study. We also thank the referees and Sunanda and Santimay Basu for their helpful suggestions.

REFERENCES

Abramowitz, M., and I. A. Stegun (1965), Handbook o[ Mathe- matical Functions, Dover, New York.

Balsley, B. B., G. Haerendel, and R. A. Greenwald (1972), Equatorial spread-F; Recent observations and a new interpretation, J. Geophys. Res., 77, 5625-5628.

Basu, S., and S. Basu (1976), Correlated measurements of scintillations and in situ F region irregularities from OGO-6, paper presented at COSPAR Symposium on Geophysical Uses of Satellite Beacons, Boston College, Boston, Mass.


Basu, S., J. Aarons, J.P. McClure, C. La Hoz, A. Bushby, and R. F. Woodman (1977), Preliminary comparisons of VHF radar maps of F region irregularities with scintillations in the equatorial region, J. Atmos. Terr. Phys., in press.

Basu, S., and M. C. Kelley (1977), Review of equatorial scintillation phenomena in light of recent developments in the theory and measurements of equatorial irregularities, J. Atmos. Terr. Phys., in press.

Briggs, B. H., and I. A. Parkin (1963), On the variation of radio star and satellite scintillations with zenith angle, J. Atmos. Terr. Phys., 25, 339-365.

Briggs, B. H., and G. J. Phillips (1950), A study of the horizontal irregularities of the ionosphere, Proc. Phys. Soc. B, 63, 907-923.

Budden, K. G. (1965), The amplitude fluctuations of the radio wave scattered from a thick ionospheric layer with weak irregularities, J. Atmos. Terr. Phys., 27, 155-172.

Chaturvedi, P. K., and P. K. Kaw (1975), Steady state finite amplitude Rayleigh-Taylor modes in spread-F, Geophys. Res. Lett., 2, 381-383.

Chaturvedi, P. K., and P. K. Kaw (1976), An interpretation for the power spectrum of spread-F irregularities, J. Geophys. Res., 81, 3257-3260.

Costa, E., and M. C. Kelley (1976), Calculations of equatorial scintillations at VHF and gigahertz frequencies based on a new model of the disturbed equatorial ionosphere, Geophys. Res. Lett., 3, 677-680.

Craft, H. D., and L. H. Westerlund (1972), Scintillation at 4 and 6 GHz caused by the ionosphere, AIAA Paper 72-179, American Institute of Aeronautics and Astronautics Library, 750 Third Ave., New York.

Cronyn, W. M. (1970), The analysis of radio scattering and space probe observations of small scale structure in the interplanetary medium, Astrophys. J., 161, 755-763.

Dungey, J. W. (1956), Convective diffusion in the equatorial F region, J. Atmos. Terr. Phys., 9, 304-310.

Dyson, P. L., J.P. McClure, and W. B. Hanson (1974), In situ measurements of the spectral characteristic of F region ionospheric irregularities, J. Geophys., Res. 79, 1497-1502.

Fredricks, R. W., and F. V. Coroniti (1976), Ambiguities in the deduction of rest frame fluctuation spectra from spectra computed in moving frames, J. Geophys. Res., 81, 5591-5595.

Fried, B. D., and S. D. Conte (1961), The Plasma Dispersion Function, Academic, New York.

Gradshteyn, I. S., and I. W. Ryzhik (1965), Tables oj • Integrals, Series and Products, Academic, New York. •

Haerendel, G. (1974), Theory of equatorial spread-F, Max- Planck-Institut fiir Physik und Astrophysik, Garching bei Miinchen, Federal Republic of Germany.

Hudson, M. K., and C. F. Kennel (1975), Linear theory of spread-F, J. Geophys. Res., 80, 4581-4590.

Kelley, M. C. (1972), Relationship between electrostatic turbulence and spread-F, J. Geophys. Res., 77, 1327-1329.

Kelley, M. C., and F. S. Mozer (1972), A satellite survey of vector electric fields in the ionosphere at frequencies of 10 to 500 Hertz, 3, Low frequency equatorial emission and their relationship to ionospheric turbulence, J. Geophys. Res., 77, 4183-4189.

Kelley, M. C., G. Haerendel, B. B. Balsley, and F. S. Mozer

(1976a), Evidence for a Rayleigh-Taylor type instability and upwelling of depleted density regions during equatorial spread-F, Geophys. Res. Lett., 3, 448-450,

Kelley, M. C., F. S. Mozer, and C. W. Carlson (1976b), Production and distribution of F region irregularities (abstract), EOS Trans. AGU, 57, 297.

Kelley, M. C., W. E. Swartz, Y. Tayan, and R. Torbert (1977), On the relationship between the plasma density profile measured in the equatorial E and F regions and simultaneous energetic particle and spread-F observations, J. Atmos. Terr. Phys., in press.

Koster, J. R. (1963), Some measurements of the irregularities giving rise to radio-star scintillations at the equator, J. Geophys. Res., 68, 2579-2590.

Koster, J. R., (1966), Ionsopheric studies using the tracking beacon on the "Early Bird" synchronous satellite, Ann. Geophys., 22, 435-439.

Lovelace, R. E., E. E. Salpeter, L. E. Sharp, and D. E. Harris (1970), Analysis of observations of interplanetary scintillations, Astrophys. J., 159, 1047-1055.

Morse, F. A., B.C. Edgar, H. C. Koons, C. J. Rice, W. J. Heikkila, J. H. Hoffman, B. A. Tinsley, J. D. Winningham. A. B. Christensen, R. F. Woodman, J. Pomalaza, and N. R. Teixeira (1976), Rep. SAMSO-TR-76-158, Aerospace Corporation, El Segundo, California.

Phelps, A.D. R., and R. C. Sagalyn (1976), Plasma density irregularities in the high-latitude topside ionosphere, J. Geophys. Res., 81, 515-523.

Piddington, J. H. (1964), Irregularities in the upper ionosphere, Planet. Space Sci., 12, 127-136.

Rufenach, C. L. (1972), Power law wavenumber spectrum deduced from ionospheric scintillation observations, J. Geophys. Res., 77, 4761-4772.

Rufenach, C. L. (1974), Wavelength dependence of radio scintillation: Ionospheric and interplanetary irregularities, J. Geophys. Res., 79, 1562-1566.

Rufenach, C. L. (1975a), Ionospheric scintillation by a random phase screen; Spectral approach, Radio Sci., 10, 155-165.

Rufenach, C. L. (1975b), Power spectra of large scintillation signals, J. Atmos. Terr. Phys., 37, 569-572.

Salpeter, E. E. (1967), Interplanetary scintillation, 1, Theory, Astrophys. J., 147, 433-448.

Scannapieco, A. J., and S. L. Ossakow (1976), Nonlinear equatorial spread-F, Geophys. Res. Lett., 3, 451-454.

Seyler, C. E., Jr., Y. Salo, D. Montgomery, and G. Knorr (1975), Two-dimensional turbulence in inviscid fluids or guid- ing center plasmas, Phys. Fluids, 18, 803-813.

Singleton, D. G. (1970), The effect of irregularity shape on radio star and satellite scintillations, J. Atmos. Terr. Phys., 32, 315-343.

Taur, R. R. (1973), Ionospheric scintillation at 4 and 6 GHz, COMSAT Tech. Rev., 3, 145-163.

Woodman, R. F., and C. La Hoz (1976), Radar observations of F region equatorial irregularities, J. Geophys. Res., 81, 5447-5466.

Yeh, K. C., and C. H. Liu (1975), Some results of scintillation studies, in Proceedings o[ Symposium on the E[[ect o[ the Ionosphere on Space Systems and Communications, edited by J. M. Goodman, Naval Research Laboratory, Washington, DC.

Ionospheric scintillation calculations based on in situ irregularity spectra

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