the two-frequency coherence function for the fluctuating ionosphere: narrowband pulse propagation

Pergamon Journal of Almosphm andSolar-Terresrrzal Physics, Vol. 59, No. 14, pp. 1831-1841, 1997

0 1997 Elsevier Scmm Ltd

PII: S1364-6826(97)000114 All rights reserved. Printed in Great Br~tan

1364-6826/97 $17.00+0.00

The two-frequency coherence function for the fluctuating ionosphere: narrowband pulse propagation

Vadim E. Gherm,’ Nikolay N. Zernov,’ Bengt Lundborg’** and Anders V&berg’

‘Institute of Radiophysics, University of St. Petersburg, Ulyanovskaya 1, Petrodvorets, 198904, St. Petersburg, Russia and ‘Swedish Institute of Space Physics, Uppsala Division, S-755 91,

Uppsala, Sweden (e-mail: [email protected])

(Received 8 July 1996; revised 10 January 1997; accepted 13 January 1997)

Abstract-This article presents a detailed numerical investigation of the two-frequency coherence function for HF wave propagation in the fluctuating ionosphere. Earlier published expressions for the coherence function are transformed to a form suitable for numerical calculation and employed in a study with an empirical plane stratified background ionosphere and a power-law spectrum of the electron density fluctuations. Our results show that the coherence function in general has a rather complicated behaviour and that it does not fall off quickly with increasing difference frequency. Under these circumstances, the case of wideband pulses require particular considerations which are presented in the companion article. The case of narrowband pulses is considered in detail and in that case the fluctuations have some influence on the pulse distortion, especially compared with the regular dispersion caused by the smooth background. They cause pulse stretching and sometimes affect the pulse amplitude through diffraction. 0 1997 Elsevier Science Ltd

1. INTRODUCTION

The present article is one in a series devoted to analytical and numerical investigations of HF pulse propagation in the fluctuating ionosphere. In a previous article (Zernov and Lundborg, 1995) we gave a review of results obtained by other authors in the area of HF pulse propagation and the influence of electron density fluctuations. We will not repeat that review here but refer to that article for background. We would only like to point out the recent article by Fridman et al. (1995) which appeared at the same time as Zernov and Lundborg (1995) and will be addressed later. In the latter article we presented general expressions describing the corruption of the pulse shape caused by the ionospheric dispersion, which results from the joint effects of the regular and the random properties of the ionosphere. To achieve this, we had to account for the influence of diffraction effects on local stochastic inhomogeneities as well as the regular refraction in the smoothly inhomogeneous background ionosphere.

In the present article, we first carry out a detailed numerical analysis of the crucial quantity in the theory

*Now at: Defence Research Establishment, P.O. Box 1165, S581 Linkoping, Sweden.

of pulse propagation in fluctuating media-the two- frequency coherence function. On the basis of this analysis, we then perform a quantitative investigation of narrowband pulse propagation in the fluctuating ionosphere with the ionospheric background profile corresponding to a given geophysical path and with physically realistic characteristics of the electron density fluctuations. These numerical calculations are fully 3D. It should also be noted that, as an inter- mediate step in our calculations, we obtain the phase and level fluctuations of the signal considered, in Gherm and Zernov (1995). The investigation of the more complicated case of wideband pulses is given in a companion paper (Gherm et al., 1997).

2. MEAN ENERGY OF THE PULSE

As before, we consider a plane-stratified model of the background ionosphere and describe the wave propagation in the isotropic approximation, i.e., any field component E,,, of angular frequency w is a solu- tion to the equation

V*E,,,+k’[~,(z,w)+~(r,w, t)]E,, = 6(r). (1)

Here Dirac’s function 6(r) represents the point source at the origin, k is the vacuum wave number, E,,(z,w) is the model of the plane-stratified back-

1831

1832 V. E. Gherm et al.

ground, and e(r,co, t) represents the local random inhomogeneities of the ionosphere with possible slow time-dependence in the quasistationary approximation for dispersive media. As a generalization from Zernov and Lundborg (1995), we consider the fully 3D case of the fluctuations so that r = (x,y,z); this makes it possible to obtain numerical results for realistic geophysical conditions. However, for prac- tical reasons we let the reference ray (the ray through the unperturbed background) propagate in the xz- plane.

In the following, we study the mean energy of the field E from (1) i.e.,

txi tee W(r, t) = (HZ*) =

I s dw,dM’(w)f’*(w)

-0c -cc

Here P(w) is the spectrum of the emitted signal and k, = w,/c, k, = WJC, with c being the vacuum velocity of light. The integral contains the undisturbed field E,“) in the plane-stratified background, represented by the main term of its ray expansion (the reference ray) in the form:

E?(r) = fo(z, w) exp{rkMr, 4w)l}, where, from Gherm and Zernov (1995),

(3)

fo(z,o) = -~ldetL”(z,0,~,~~)I-“2[~O(z)-~~-_~]~’i4

x [E”(O) - a; - pi] - “4, (4)

M, Q) = a0x + BOv + Uz, 0, ~0, PO)

+ T sgn L”(z, 0, cto, Pd.

The quantities ~~ and /IO are the direction cosines with the x and y axes of the ray direction at the source. Since the propagation of the reference ray is in the plane y = 0, we have /IO = 0; however, when calculating the derivatives of L below it is necessary to retain the /$,-dependence. The implicit dependence of t( on w in these expressions is determined by the ray equations for the ray from the origin through the point (_~,y,z):

x + L,(z, 0, %I> PO) = 0,

Y + &(z, 0, ao, PO) = 0, (5)

where subscripts on L denote partial derivatives with respect to an angular variable. The phase function L is of the WKB type; for arbitrary z, z’ it has the form

L(z,z',a,b) = I

‘Jwdz, (6) I’

if there is no reflection in the ionosphere or if the wave has not reached the reflection level, and:

L(z, z’, a, P) = s

‘Jmdz i’

after reflection at the level z” given by the condition

&o(Z) - c? - /I’ = 0. (8)

The phase shift 7c/2 in (7) is caused by the reflection. We may form a matrix L” from the second deriva-

tives of L with respect to the angular variables. Its determinant is given by

det L”(z, z’, CC, /?) = L,,L,, - L,,L,,,

sgn L”(z, z’, c(, 8) = v+ -v_, (9)

with V+(X) being the number of positive (negative) eigenvalues of L”.

The mean energy of the field, given by (2) also contains the two-frequency coherence function T(r, w,,w,). In the case of quasistationary (slow) motion of the ionospheric random inhomogeneities this coherence function can actually also be a function of the slow time variables t, and t2. If stationarity is assumed for the random process this dependence will be of the form t, - tZ. When considering the mean energy of a pulse, as in (2) one will then obviously need the quantity T(r,w,, w,,O). Below we omit the last zero-value time argument of the two-frequency, time coherence function. In the following section, we briefly discuss how to obtain its representation.

2.1. Coherence function

As considered in Zernov and Lundborg (1995) we use here Rytov’s method generalized to the case of an essentially inhomogeneous background medium with local embedded electron density inhomogeneities (Zernov, 1980). The method involves a complex phase $ which accounts for the influence of the local inhomogeneities on the field so that

&Jr, 0 = E<,“‘(r) exp [ti(r, 01. (10)

The procedure for obtaining the complex phase in the forward scattering approximation can be found in earlier articles (Zernov, 1980; Zernov and Lundborg, 1995; Gherm and Zernov, 1995). The essence of the method is that tj can be written as a perturbation series:

HF pulse propagation through a fluctuating ionosphere: narrowband 1833

Il/(r, 4 = l(ldr, 4 + h(r, 4 + , (11)

where the different contributions can all be expressed in the same form:

t+b(“)(r, t) = [EE)(r)]m ’ jG(r, r’)fm(r’, t)El)O)(r’) dr’,

(12)

with fm(r’, t) being known functions. Below, we will need $, = I#‘), $z = I++(‘), and (rj2) = tiC3’, and for them we use

h = -Wr, 0, .h = -(V$,Y, f3 = -(WI)‘>. (13)

To analyze each of the expressions (12), it is convenient (Zernov, 1980; Gherm and Zernov, 1995) to introduce ray-centred (local ray) variables linked to a reference ray which connects the source and the point of observation in the undisturbed medium; cf. Figure 1 of (Zernov and Lundborg, 1995). Using these variables with s calculated along the ray, n perpendicular to the ray in the plane of propagation, and r perpendicular to the plane of propagation, each of the expressions (12) can be written in the Fresnel approximation for forward scattering as follows

+ sgn (R(s’))l (14) I

Here

D, ‘(3’) = - [L,‘(z’, 0) + L,‘(z, z’)] = &o(d) -a; ’

D; I($‘) = - [LpB’(z’, 0) + Lgp’(z,z’)], (15)

and E,,(S) must be understood as E&, 0, 0), i.e., s0 given at points on the reference ray. To obtain (14), we used the representation (3) for the incident field I?$‘) and a corresponding expression for the Green’s function G(r, r’). The integration is to be carried out over the main Fresnel volume from the source to the point of observation.

The particular cases of (14) needed in the following are, with (13)

$,(r, t) = g sslds’ dn’ dt’ &(S’, n’, r’)

&~“(S’)lO,(S’)O,(S’)I Ii2

+ sgn (U.O)l , (16) I

+ sgn (Ws’))l (17)

The expression for (ti2) is simply (17) with (V$,)’ replaced by ((V$,)‘>.

By this, we have expressed all the quantities needed for constructing the coherence function appearing in equation (2) for the mean energy of a pulse which has passed through the fluctuating ionosphere. Using the first and second approximations of the complex phase, I/I, = x, + is, and tj? = x2+ is,, this function hence takes the following form:

f = T2(r, ml, ~2) = exp Mr, aI, 0,)-t iB(r, wI, wJ1,

B(w, a = <Mw)) - (UwJ) + <[x,(d +xdw)I[~,(w- WGD- (18)

As one can see, these expressions contain different moments of the real and imaginary parts of the complex phases rj, and ti2. Among them are the single- frequency moments (average values and the variances for the frequencies w, and CL)J as well as the two-point moments (two-frequency autocorrelation and cross- correlation functions). For further consideration it is convenient to separate the single-frequency and two- frequency moments, writing the coherence function in the form of the following product:

f2(rr ml, oz) = VJr, 0,) V,*(r, WJ

xexp[b(r,wl,w,)+ig(r,wl,wz)l, (19) where,

1834 V. E. Ghenn et al.

The coherence function, written in this form, will be further investigated below and calculated for specific model cases. To this end we shall need more convenient expressions for the various moments. These are also given in the next section.

In a previous article (Zernov and Lundborg, 1995) we restricted the treatment to cases where the real and imaginary parts A and B of the argument of the exponential function in f2 could be expressed as series in the frequency variables, accounting only for the terms up to the second order. The detailed numerical calculations of the ‘exact’ coherence function to be presented below will show that this approximation is sometimes insufficient. However, it will also be seen that it is a valid description for narrowband pulses in the fluctuating ionosphere.

2.2. Construction of the moments

To calculate the ‘exact’ coherence function, we need various second-order moments of the first-order terms and the average of the second order terms of the perturbation series for the complex phase of the field; see (20). For the further treatment it is more convenient to introduce local Fourier-conjugate variables k,, k,, k, for every point on the reference ray, and to represent the fluctuations in the spectral domain. In this case, we obtain for the first-order term:

correlation function for the dielectric permittivity fluctuations.

For the two-frequency moments it should be taken into account that different carrier frequencies cor- respond to different rays reaching the point of observation:

($,(w,)$f(w )) = *lr” ds 2

4 0 [Eo(S&O(S~“~

*exp{-w[ ]} J@,(S) + K&(S) (23)

Here k, and k2 are the vacuum wave numbers of the two frequencies. The integration over s is performed along the ray for the mean frequency (w, +w,)/2 and the perpendicular through the point s crosses the neighbouring rays for w, and w2 at the points with coordinates S, and s2, respectively. A(s) is the distance between the points s, and s2.

As for the average (&) of the second-order term, we can construct it in a similar way. Hence we obtain V$, from (21) and subsequently

exp [K$;(s’, s) + K$:(S’, s)] (24)

* exp and then, finally,

(2 1)

where E(K,, K,, K,) is the spectrum of the fluctuations and R,(s’) and R,(s’) are the projections of the vector r

* exp K:D:(S',S)] (25) I

on the axes of the local ray-centred coordinate system. The procedure for constructing the second-order

These are single-frequency expressions but it should

single-frequency moments of complex phase from (21) be noted that DL(s’, s) and D:(s’, s), which are obvious

is almost the same as in Gherm and Zernov (1995). generalisations of (15) actually depend on two vari-

Thus, for the quantity ($:(a~)) we have ables s and s, along the ray:

($:(w)> = - ; ‘s s * sj dK,I dK,B,(O, Km K,) 0 0

. exp i

- f [r@,(S) + K?D,(S)I . (22)

where B,(O, k,,,k,) is the transverse spectrum of the

Dh(s’, s) = - I!&‘, O)L,,(s, s’) E&‘) - ctx:,

&(s’, 0) + &z(& s’) Eob’) ’

Di(s’, s) = - &&‘, O)&,ds, 0

L,,p(.f, 0) + L,&, .f) (26)

As one can see from equations (22) (23) (25) for the moments involved in the description of the quantities V, h, 9, which form the coherence function


according to formula (19) these moments are in general all expressed in the form of rather complicated multiple integrals. A representation of the coherence function similar to that given by our equation (19) has also been considered in Fridman et al. (1995). Except for the absence of any specification of the function V in that article, our representation differs from theirs also by the presence of the non-zero imaginary part q in the exponent of our equation (19). Another difference is that the quantity b, the real part of the function in the exponent in (19) is expressed in the cited article as a single integral. Strictly speaking, this is not correct. As one can show by considering the general representation for the function b + iq in equation (23) the situation as given in Fridman et al. (1995) i.e. q = 0 and h represented as a single integral, does not apply to the general case when diffraction effects have to be taken into account. It is formally true only in the geometrical optics approximation. The limiting transformation to the geometrical optics case can be performed by putting the argument of the last exponent in equation (23) equal to zero. In this particular case q becomes zero and the integrals over the spectral variables k,, k, in equation (23) can be calculated explicitly to give the following result:

where K,[A(s), 0] is the correlation ionospheric fluctuations integrated tudinal variable s.

function of the over the longi-

In the general case the multiple integrals in the representations (22), (23) (25) have to be calculated

(27)

as they are. However, it follows from the general relationship (23) for b+iq that the approximation considered in Fridman et al. (1995) is quite adequate in the case of small values of the difference frequency 6.

The goal of the present article, as well as the companion article (Gherm et al., 1997) is to give the general description of the two-frequency coherence function for a wide range of frequencies in order to describe even wideband pulse propagation in the fluctuating ionosphere. Besides, as our analysis shows, the behaviour of the mean field gives an essential contribution to the coherence function even in the case of zero value of the difference frequency 6 (this is an effect of the mean energy redistribution which is exclusively caused by diffraction of the HF field on the local random inhomogeneities of the ionosphere); see equation (15) of Gherm et al. (1997). The mean

field also determines the behaviour of the coherence function for large values of the difference frequency 6. Therefore this dependence also has to be taken into account for a correct description of the coherence function. In calculating this function the first and the second approximations of the complex phase must hence be involved according to (20).

All this shows the necessity to use the most general representation of the coherence function, as given by (19), (20) in the numerical calculations. Above we have called this the ‘exact’ coherence function. Below we discuss the details of the numerical calculations of this function and give results of numerical simulations for realistic models of the background ionosphere and the ionospheric electron density fluctuations.

2.3. Model of thejluctuations

For the concrete calculations, we now need the function B,(K), the correlation function spectrum of the fluctuations. We shall here adopt the isotropic power-law spectrum model:

where I is now the traditional gamma function; o’, is the variance of the fractional electron density fluctuations, which is considered as a homogeneous zero- mean random field; k, = 27c//,, where d, is the outer scale size; and p is the spectral index. Slow spatial variations of the spectral parameters can easily be accounted for by specifying the functional depen- dencies k, = k,(s), p = p(s) and cN = CJ&).

2.4. Numerical calculations

Below we discuss in some detail the numerical evaluation of the coherence function as given by (19,20). As was already pointed out, some expansions of the quantities A and B were given in the previous article (Zernov and Lundborg, 1995). Our following analysis will reveal the validity of these expansions and how they can be employed in quantitative calculations.

The expressions (22, 23, 25) which are the values to be calculated numerically, are multiple integrals consisting of double integrals in the spectral domain and single or double integrals along the ray path. The direct numerical integration of these expressions is quite a time-consuming procedure. For isotropic inhomogeneities one integration in the spectral variables can be performed explicitly by introducing polar

1836 V. E. Ghenn et al.

instead of Cartesian coordinates in the wave vector integration domain. After that the integration over the polar angle in the single frequency expressions (22,25) can be performed analytically, resulting in expressions involving the zeroth order Bessel function J,:

where K~ = fcf, + K:. The integrations over s and s’ are performed along the reference ray which connects the transmitter and the receiver.

For the case of the two-frequency moment (23) the same procedure gives the more complicated expression:

II (30)

In fact, our calculations have shown that the sum in this formula gives only a very small contribution in the integral in comparison with the term consisting of zero order Bessel functions, and can be neglected.

As a model of the background ionosphere in our calculations, we have chosen a summer daytime profile for a rather high latitude with an E and an F layer. The propagation distance was 900 km and the frequency range investigated is 7.5-8.5 MHz. Instead of considering two separate frequencies w, and w2, we have introduced the centre frequency R = (0, +a*)/2 and the difference frequency 6 = w, -w2 as new variables. Hence we present our results as 3D plots of the

modulus and the phase of the coherence function as functions of s1 and S in the above-mentioned frequency band. For convenience we indicate in the figures the wave frequencies corresponding to the angular frequencies R and 6. All the calculations were done to a controlled accuracy of 1%.

Figures 1 and 2 show 3D plots of iI-1 for two values of the outer scale: /, = 10 km and 3 km. They clearly have similar shapes and some common features, such as a maximum when 6 = 0 and a tendency to fall off with increasing 6. The width of the peak in the 6 direction depends on the outer scale size, the higher L, the greater the width. We also see that iI-1 is slowly varying with the centre frequency R. Figures 3 and 4, show some sections taken from Figs 1 and 2, viz., 2D plots of iI-1 as a function of the difference frequency for a few fixed centre frequencies.

The function B(R, 6) in (1 S), the phase of the coherence function, is plotted in Fig. 5. It is equal to zero when 6 = 0 and varies monotonically with 6; its derivative with respect to 6 at the point 6 = 0 depends on the central frequency Q and in any case the phase B is small. Hence the expansion (38b) in Zernov and Lundborg (1995) can be used for sufficiently small 6 with the coefficient p,(Q) slowly dependent on D.

At first glance, it seems that the coherence function must tend to unity when the difference frequency 6 tends to zero, since in this case of coinciding argu- ments of the function it simply must give the mean energy of the monochromatic field. However, there is no reason to believe that this mean energy in the medium with fluctuations should be the same as in the medium without fluctuations. On the contrary, our calculations show that in the case when diffraction effects are essential one finds some redistribution of energy caused by this diffraction. Hence the spatial location of the mean energy in the medium with fluctuations is different from the location in the undisturbed medium. In the 3D plots of the coherence function one can see a slow dependence on the centre frequency when the difference frequency is put equal to zero. The stronger the diffraction effects, the more pronounced is this dependence.

The main conclusions of the above are the following:

1. The modulus of the coherence function has a maximum when the difference frequency is equal to zero.

2. The width of the maximum depends on the outer scale of the permittivity fluctuations spectrum, the greater the scale the greater the width.

3. The maximum value of the coherence function depends slowly on the centre frequency.


6 [MHz] 0.3 7.5

-8.5

Fig. 1. The modulus of the coherence function, Ir1, as a function of the centre and difference frequencies R and 6. The magnitude of the fluctuations is CT; = 5. 10m6 and the spectral index isp = 3.7. The outer scale

/<, is here 10 km.

0.8

0.6

Fig. 2. Same as Fig. 1, but the outer scale L,. is 3 km.

-8.5

4. The modulus of the coherence function tends to This discussion of the numerical results summarizes non-zero limiting values as the difference frequency the general behaviour of the coherence function. It is increases. now clear that the analytical technique developed in

1838

0.3

P-I 0.25

V. E. Gherm et al.

e, = 10

-- 0.05 0.1 0.15 0.2 0.25 0.3

b [MHZ] Fig. 3. The modulus of the coherence function, jrl, as a function of the difference frequency 6 for three

fixed centre frequencies 51. The values of oN and p are as in Fig. 1 and the outer scale is 10 km.

0.9

0.85

0.8

0.75

0.7

0.65

0.6

R = 8.5 ,-

-

I-

0.45 I- I I I

O 0.05 0.1 .-

0.15 0.2 0.25 0.3

0.55

0.5

I I

ec = 3

6 [MHz] Fig. 4. Same as Fig. 3, but the outer scale is 3 km.

HF pulse propagation through a fluctuating ionosphere: narrowband

B o

-0.02

-0.06

-0.08 8.5

Fig. 5. The function B(QJ), i.e. the phase of the coherence function. The magnitude of the fluctuations is Si = 5. 10m6 and the spectral index is p = 3.7. The outer scale e, is 3 km.

Zernov and Lundborg (1995) for describing wideband pulse propagation must be slightly modified to account for the essentially nonzero character of the coherence function for large values of w, - w2. This can be done, but needs some further detailed considerations which will be reported in a follow-up article by the present authors (Gherm et al., 1997). Hence we will give in that article the detailed investigation of wideband pulse propagation in the fluctuating ionosphere, on the basis of the above-mentioned modified analytical technique and the numerical investigation of the coherence function given in the present article. To finish the present investigation we will now present results of quantitative calculations of the influence of the regular as well as random dispersion of the ionosphere on narrow-band pulse propagation.

3. NARROWBAND PULSE PROPAGATION

As in Zernov and Lundborg (1995), we consider an emitted narrowband Gaussian pulse of duration To and centre frequency w,, which has the spectrum

P(W) = ~ Bexp L

- +UJj (31)

The received energy for this pulse is according to (65-67) in Zernov and Lundborg (1995) given by the following analytical expression:

(32)

with

(33)

The representation (32) was obtained for a quadratic phase model of the transfer phase function of the background ionospheric channel (Lundborg, 1990). The quantity b,, in this representation is not to be confused with /I,, in Section 2.

To describe the coherence function we use here the parameters PO(o), b,(o) and /$(w) which define the narrow central parabolic peak of the exact coherence function; they are the coefficients of a Taylor expansion of A and B in (18); see (38a,b) in Zernov and Lundborg (1995). The quantity j&(w,) is defined by the least squares method to provide the best fit of the numerically calculated coherence function to a Gaussian shape of the narrow central peak. The notion of a narrowband pulse will then be defined

1840 V. E. Gherm et al.

relative to the width of the exact coherence function by the inequality

To > 2&j&% (34)

i.e., the opposite of equation (39) in Zernov and Lund- borg (1995). This means that the domain of nonzero values of the pulse spectrum is smaller than the peak part of the exact coherence function, Consequently, the main contribution to the integral describing the pulse mean energy is actually given by an area of the order of the pulse spectrum width. So, the inequality (34) states the validity of the expressions (32) and (33) for the pulse shape and also gives the definition of the notion of ‘narrowband signal’.

Below, we perform some exercises to demonstrate the contributions of the different kinds of dispersion to the narrowband pulse stretching. We perform the evaluation for a carrier frequency w, of 8 MHz and the outer scale of the fluctuations d, = 3 km. The model of the peak part of the coherence function is given by (38a,b) in Zernov and Lundborg (1995). With our parameters, the modulus of the peak is then given by

A = a;(-4.77 x 104-2.37x 106d2), (35)

where 6 is the difference angular frequency w, - w2 in MHz. Different researchers choose as the variance of the relative electron density fluctuations, a&, values of the order 10e4 to 10m6; in our calculations we have chosen & = 5. 10m6, so that we have here the case of Figs 2,4 and 5. Then we get

PO = -0.24, 2s = 7~s. (36)

This means that, with the parameters chosen here, we have a narrowband pulse with regard to the propagation if, according to (34) its duration is longer than 7~s i.e. the spectrum width estimated as Af= (47~&)-’ should be essentially less than 23 kHz. For the chosen electron density profile and 8 MHz carrier frequency, we also have, because of the regular dispersion,

$[/&,(w,)] = 2 x 10-5(ms)2. (37)

The expression (32) gives the following result for the duration of the narrowband pulse after stretching caused by the passage through the disturbed ionosphere:

T = To 1 +4T,y2~,(w,)+ T,y4 (

$5 LkM~c)l)

(38)

Let us evaluate the contributions of the second and

third terms under the square root for the duration T,, = 0.1 ms of the launched pulse. This satisfies the condition (34) pretty well. Then, finally, the stretched duration according to (38) is;

T = 0.141+5 x lO-3+4x 1O-6

= 0.1(1+2.5x 10e3)ms. (39)

This shows that the influence of regular dispersion [through (37)] is negligible, as it must be for narrowband wave packet propagation, and that the dispersion resulting from the fluctuations forces a stretching of the order of less than 1%.

Additional effects caused by the fluctuations are the slight change in the amplitude of the pulse because of the factor exp [/I&w,.)] z 0.79, essentially the result of diffraction effects on the fluctuations, and the additional group delay time caused by the term /3,(w,) in (33). The latter can be found from Fig. 5 and gives a value of 0.33~s for the carrier frequency 8MHz. This is a much smaller value than both the group delay 3.8 ms and the transmitted pulse length 0.1 ms. Therefore the influence of the fluctuations on the group delay is negligible.

4. CONCLUSIONS

We have investigated in detail the general properties of the two-frequency coherence function, which is fun- damental in calculating pulse propagation through the fluctuating ionosphere. In this study we have considered real ionospheric background parameters and accounted for diffraction effects caused by the ionospheric electron density fluctuations.

We found that the coherence function in the general wideband case does not fall off rapidly enough as a function of the difference frequency. This brings additional difficulties into the pulse energy calculations, so that some more general analytical representations, obtained in Zernov and Lundborg (1995), must be used accurately and sometimes also need some modifications. That is the subject of the companion paper to the present article (Gherm et al., 1997).

As for the narrowband pulse propagation, it is well described by the technique developed in Zernov and Lundborg (1995). The main result for this case is that the dispersive property of the regular background ionosphere only produces a group delay in the propagation and practically no pulse shape distortion. The dispersion caused by the fluctuations gives rise to pulse stretching which is of the order of some percent of the initial pulse length. It also affects the pulse amplitude and gives a small shift of the group delay time.


Acknowledgements-The results described in this publication were partly made possible by the financial support of RFBR, Grant No 96-02-17182, and of the international program ‘INTERGEOFIZIKA’ of St. Petersburg State University.

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