STATISTICAL NATURE OF THE IONOSPHERIC STRUCTURE

Ya. L. Al’pert

§ 1. INTRODUCTION

Recently, in investigations of the ionosphere, a new direction has emerged, to which quite a number of works have already been devoted. The essence of these investigations and their novelty lie in the statistical method of analyzing the variability of the state of the electromagnetic field of an individual radio signal reflected from the ionosphere, and in studying, on this basis, its fine structure.

The variability of the state of the electromagnetic field of radio waves at the receiving point, caused by the influence of the ionosphere, has long attracted the attention of a number of investigators. There are many works in which different types of so-called fadings, or fading of radio waves, and the causes by which they are conditioned, have been studied (see, for example,¹, ², ³). However, in most of these investigations, especially experimental ones, the phenomena causing variability of the state of the field appeared in their aggregate form. Usually such experimental data were considered in which the different components of the wave reflected from the ionosphere were not separated individually, namely the ordinary and extraordinary waves conditioned by the double refraction of the ionosphere. In addition, often in these experiments the field at the observation point consisted of a wave arriving along the earth’s surface and of waves of different orders reflected from the ionosphere. Therefore, owing to the difference and inconstancy of the propagation paths, the propagation velocities, the reflecting power from the earth, absorption, the state of polarization, and other parameters of each of the components of the field, the resultant field had a complex character, and it was extremely difficult, if at all possible, to isolate and especially quantitatively investigate the different causes producing its instability. Naturally, in these cases the variability of the field strength often had to be chaotic. Indeed, even if each of the phenomena indicated above proceeds with some regular law, then, owing to the absence of a constant

connections between them and the multitude of these phenomena, their cumulative action must have a more or less random character. And since the nature of these phenomena is such that many of them proceed nonuniformly and often are themselves chaotic in time (see below), it is even more understandable that, with the corresponding processing of experimental data, it was found that the field strength of the received wave had a Gaussian distribution².

At the same time, the formulation of such experimental investigations and such a methodology for processing them are of great interest, since they would make it possible, on the basis of an analysis of the variability of the parameters of the wave reflected from the ionosphere, to study various phenomena that cause this nonconstancy. Certain possibilities for a more detailed investigation of one of the causes of the variability of the field of the reflected wave are provided precisely by statistical methods of analyzing a single signal reflected from the ionosphere. By the singleness of a signal is meant not only that it is one signal reflected from one region of the ionosphere, but also that this signal is one magnetically split component—ordinary or extraordinary wave. This condition is mandatory, since otherwise the observed changes in the amplitude of the signal will be a consequence not only of changes in the conditions in the reflecting region, but also the result of interference of these two waves, i.e. of the so-called polarization fading, and the data on the structure of the reflecting region of the ionosphere obtained in such experiments may often be erroneous. This circumstance must be stipulated, since a number of experiments carried out recently, in which, on the basis of statistical methods of processing them, structural features of the ionosphere are elucidated, do not guarantee such purity of the measurements (for example, in work⁴), and therefore the conclusions from these works are still not sufficiently reliable. Obtaining experimental data in the sense specified above is possible, for example, either by using so-called polarization receiving antennas⁵, by means of which one component of the wave is isolated, or by using time selection, i.e. when both signals (ordinary and extraordinary), resolved in time, are present on the oscillogram, and during rapid cinematographic recording the changes in the amplitudes of both signals are observed⁶ (see, for example, Figs. 1 and 2).

It had already been pointed out in earlier works that one of the causes of the variability of the field of a wave reflected from the ionosphere must be the inhomogeneity of the reflecting layer of the ionosphere, i.e. the presence in it of so-called ionized clouds (see, for example,⁷,⁸). In later works, on this same basis, the causes of amplitude fluctuations of reflected signals were analyzed theoretically (see, for example,⁹), and the results of experimental investigations were processed, with the aim, in particular, of clarifying the character י

of the influence of these inhomogeneities10, 11. Numerous investigations of the ionosphere have established that the ionospheric layers are indeed inhomogeneous in their structure and that sporadic layers of cloud-like structure often appear in them, while during periods of strong disturbances the structure of the layers, especially of the higher layer \(F\), becomes patchy (see, for example,12).

Fig. 1. Photograph of pulses reflected from the ionosphere, taken from an oscilloscope screen. In the lower part, along the time line, the emitted signal is visible, as are its doublets (ordinary and extraordinary) reflected once, twice, and three times from the ionosphere. In addition, time marks are visible—at every 100 km of layer height. In the upper part of the figure, one doublet of signals is shown separately on a larger scale.

However, the following question naturally arises, which is of primary interest within the scope of this article: what takes place in the reflecting region of the ionosphere when the ionosphere is, as it is customary to say, “quiet,” and when observations are made of a single signal and it does not “split,” does not become “diffuse” (see § 4, d), and broadens practically very little, i.e., when no signs of ionospheric “disturbance” are noticeable? Of course, it is then necessary to stipulate precisely what is to be called a “completely undisturbed” ionosphere. (Within the data considered here, this will be done in § 4, d).

Experiments have shown that, even in such a quiet ionosphere, the amplitude of the reflected signal almost always varies (often chaotically) with time and from point to point along the Earth’s surface, which indicates a complex structure of the reflecting region. This circumstance led to the assumption that “reflection” from the ionosphere has a complex character, namely, that along with

regularly reflected wave, one may say a “specularly reflected” wave, a bundle of waves arrives at the receiving point, formed by various scattering centers of the ionosphere, which

Fig. 2. Motion-picture sequence (at a rate of 6 frames per second) of a doublet of magnetically split signals reflected from the ionosphere (see the caption to Fig. 1).

are all the time in chaotic motion and which, consequently, are located within the reflecting region, i.e., that part of the ionosphere,

which forms the reflected signal \(^{6,14,13}\). Naturally, the same picture may be observed if one assumes that the rough, chaotically inhomogeneous reflecting region preserves its “shape,” but moves as a whole relative to the observation point. It has long been established that such drifts (see § 5) exist in the ionosphere and can indeed lead to variability in the state of the reflected wave. However, it follows from simple calculations (see § 4, c) that, because of diffusion, which proceeds rapidly, especially in the \(F\) layer, the “lifetime” of an inhomogeneity is very short, and during an experiment lasting several minutes there must occur a replacement of some inhomogeneities by others and a change in the structure of the reflecting region. Thus one may say that the “shape” of the reflecting region of the ionosphere changes almost continuously, and it is precisely this that determines the nonstationarity of the state of the “reflected” wave, while the rate of this process determines its character.

We have repeatedly used above the term “reflecting region,” without precisely specifying this concept. Naturally, the question arises as to the size of this region, i.e. what part of the ionosphere mainly participates in the formation of a single reflected signal? Having obtained an answer to this question, one may naturally try to determine the “sizes” of the scattering centers located inside the reflecting region, i.e. to obtain an idea of the structural features of the ionosphere. These questions are discussed in § 4, a), b), where the results of corresponding estimates of these quantities, obtained using published experimental results, are presented.

Consideration of all the questions briefly noted above, on the basis of the available experimental data, already makes it possible at present to conclude that even under conditions when the ionosphere is “completely calm,” it always consists of inhomogeneities of “small” size (“granules”)—scattering centers that are in a state of continuous motion and change. The mean linear dimensions of these inhomogeneities, the velocities of their motion, and the variability of these parameters under different conditions have not yet been studied sufficiently fully. However, different investigations yield quantities of approximately the same order. Thus these investigations reveal the character of the fine structure of the ionosphere and show that it is a “turbid” medium. The measured values of the “degree of turbidity” (see § 2, b) indicate that in some cases the intensity of the specularly reflected wave is negligibly small in comparison with the intensity of the scattered signals. Such a structure of the reflecting region, consisting of a large number of such “granules,” leads to the result that the wave “reflected” from the ionosphere is a beam—a cone of waves arriving from different directions and grouping זיך around

regularly reflected wave, one may say a “specularly reflected” wave, a bundle of waves arrives at the receiving point, formed by different scattering centers of the ionosphere, which

Fig. 2. Motion-picture recording (at a rate of 6 frames per second) of a doublet of magnetically split signals reflected from the ionosphere (see the caption to Fig. 1).

are all the time in chaotic motion, and which, consequently, are located inside the reflecting region, i.e., that part of the ionosphere,

near some mean direction. In § 3, b) a method is described for determining the “angular spread” of this wave cone.

At present it is impossible to draw any conclusions about the nature of these scattering inhomogeneities, i.e., it is not yet possible to say whether they constitute some isolated ionized clouds—as it is customary, for clarity, to say—arising for one reason or another as the result of many local “bursts” of ionization; whether these are wave-like processes, i.e., condensations and rarefactions of density caused by longitudinal oscillations of the plasma; or inhomogeneous formations of some other structure. Nor are there data that would make it possible to put forward any reasonably confident assumptions about the principal causes leading to such a structure of the ionosphere. Some phenomena that may lead to this are very briefly noted below (§ 6); however, the considerations given in the literature are still of a very preliminary character.

§ 2. STATISTICAL INVESTIGATIONS OF THE AMPLITUDE OF SIGNALS REFLECTED FROM THE IONOSPHERE

It was already indicated above that the character of the behavior of the amplitude of a single signal reflected from the ionosphere led to the assumption that the electromagnetic field at the observation point is a superposition of the specularly reflected wave and a group of scattered signals. Mathematically this means that the field strength \(E\) and the square of the amplitude \(R\) of the reflected wave are respectively equal to

\[ E=a_0\cos(\omega_0 t-\varphi_0)+\sum_{(s)} a_s\cos[(\omega_0+\Omega_s)t-\varphi_s] \tag{1} \]

and

\[ R^2=\left\{a_0+\sum_{(s)} a_s\cos(\Omega_s t-\varphi_s)\right\}^2 +\left\{\sum_{(s)} a_s\sin(\Omega_s t-\varphi_s)\right\}^2, \tag{2} \]

where \(\omega_0=2\pi f_0\) is the angular carrier frequency of the incident wave, while \(a_s\), \(\varphi_s\), and \(\Omega_s\) are the amplitude, phase, and frequency displacement of the scattered wave—random quantities; moreover, before the validity of (1) and (2) is clarified, no assumptions are made concerning the cause of the frequency displacement \(\Omega_s=2\pi(f_s-f_0)\) of the components \(a_s\cos(\omega_s t-\varphi_s)\).

Thus, first of all it is necessary to obtain from experimental data an answer to the question of the validity of the basic assumption expressed by formula (1). For this purpose one must compare the distribution curve, or, in other words, the probability density \(W_{\mathrm{э}}(R)\), of the amplitude \(R\) of the signal obtained from different experiments with the function \(W(R)\), computed theoretically for expression (1).

Having a series of successive values of the signal amplitude \(R_i\), measured at equal, sufficiently small time intervals \(\tau\)

Fig. 3a. Curves illustrating the character of the variability of the amplitude of an individual signal in different experiments. The points indicate the values \(R_i\), measured every \(1/6\) sec.

(see the points in Figs. 3a and 3b, on which the results of the corresponding measurements are presented), one can construct a distribution curve-

of \(W_{\mathrm e}(R)\), by dividing the values \(R_i\) into a finite number of equal intervals \(\Delta R_i\) and computing, for each interval, the quantities

\[ W_{\mathrm e}(R_i)=\frac{N_i}{\Delta R_i \sum N_i}. \tag{3} \]

where \(N_i\) is the number of signals having an amplitude in the interval \((R_i, R_i+\Delta R_i)\), and \(\sum N_i\) is the total number of all measurements of signal amplitude.

Fig. 3b. Curves illustrating the nature of the variability of the amplitude of a single signal in different experiments.

Theoretically, the probability density \(W(R)\) can be calculated by methods well developed in the theory of electrical fluctuation processes. Without dwelling on these calculations (see), we shall give only the final formula. One obtains

\[ W(R)=\frac{2R}{\sum a_s^2}\, e^{-\frac{R^2+a_0^2}{\sum a_s^2}}\cdot I_0\!\left(\frac{2a_0 R}{\sum a_s^2}\right). \tag{4} \]

where \(I_0\) is the Bessel function of zero order with imaginary argument—the so-called modified Bessel function

\[ I_0(x)=J_0(ix). \]

* The corresponding methods are set forth, for example, in the book by V. I. Buyanov, “Fluctuation Processes in Radio-Receiving Devices,” (Publishing House Sovetskoe Radio (1951)). We recommend this book to the attention of readers.

From the theoretical function (4) obtained, it is clear that in order to compare it with the experimental curve obtained by means of (3), it is necessary to know the values of \(a_0^2\) and \(\sum a_s^2\). Calculation of \(\overline{R^2}\) and \((\overline{R})^2\) makes it possible to determine them. From (4) one obtains

\[ \overline{R^2}=a_0^2+\sum a_s^2 \tag{5} \]

and

\[ \overline{R}= \frac{\sqrt{\pi \sum a_s^2}}{2}\, e^{-\frac{a_0^2}{\sum a_s^2}} \left[ \left(1+\frac{a_0^2}{\sum a_s^2}\right) I_0\left(\frac{a_0^2}{2\sum a_s^2}\right) + \frac{a_0^2}{\sum a_s^2} I_1\left(\frac{a_0^2}{2\sum a_s^2}\right) \right], \tag{6} \]

whence it follows that

\[ \frac{\overline{R^2}}{(\overline{R})^2} = \frac{4}{\pi} \frac{e^{\beta^2}(1+\beta^2)} {\left\{(1+\beta^2)I_0\left(\frac{\beta^2}{2}\right) +\beta^2 I_1\left(\frac{\beta^2}{2}\right)\right\}^2}, \tag{7} \]

where

\[ \beta^2=\frac{a_0^2}{\sum a_s^2}. \tag{8} \]

The curve of the dependence of \(\dfrac{\overline{R^2}}{(\overline{R})^2}\) on \(\beta\) is plotted in Fig. 4. Thus

Fig. 4. Dependence of \(\dfrac{\overline{R^2}}{(\overline{R})^2}\) on \(\beta\); \(\beta^2\) is equal to the ratio of the energy of the “specularly” reflected wave to the total energy of the scattered signals.

it is seen that, calculating from the experimental data the values

\[ \overline{R^2}=\frac{\sum N_i R_i^2}{\sum N_i} \quad\text{and}\quad \overline{R}=\frac{\sum N_i R_i}{\sum N_i}, \]

one can determine from the curve of Fig. 4 the value of \(\beta\); then, with the aid of (5), the quantities \(a_0^2\) and \(\sum a_s^2\), and, finally,

substituting them into (4), construct the theoretical distribution curve \(W(R)\) corresponding to the experiment under consideration. To compare the experimental curve \(W_e(R)\) with the theoretical one, it is convenient, however, to construct the corresponding curves in dimensionless coordinates, i.e., to compute the functions

\[ W(x)=\sqrt{\frac{\sum a_s^2}{2}}\cdot W(R), \tag{9} \]

where

\[ x=\sqrt{\frac{2}{\sum a_s^2}}\cdot R. \tag{10} \]

Fig. 5. Comparison of the distribution curve of the signal amplitude, computed theoretically (solid line), with the corresponding values obtained from processing experimental data: points denote values corresponding to an ordinary wave, and crosses to an extraordinary one. Along the ordinate axis are plotted the values

\[ \sqrt{\frac{\sum a_s^2}{2}}\,W(R), \]

and along the abscissa axis—the values

\[ \sqrt{\frac{2}{\sum a_s^2}}\cdot R. \]

Processing numerous experiments by the method described here showed that in most cases the results of the experiments agree well with the results of theoretical calculations. For illustration, Fig. 5 shows several cases of comparison of the theoretical curves, calculated by formula (9), with experimental values calculated from measurements by formula (3). Near the curves are written the values of \(\beta\) corresponding to the given experiment. It has been established that in a number of cases there is some discrepancy between the theoretical and experimental results, which may be a consequence of the incompleteness of the theory presented, in which, in particular, it is not taken into account that, along with the field components \(a_0\cos\omega_0 t\) and \(\sum a_s\cos(\omega_s t-\varphi_s)\), there may be, for example, a component \(a_d\cos\omega_d t\) with a stable value of \(\omega_d\). In this case it is understood that the frequency shift \(\omega_d-\omega_0\) is caused by a constant

by a drift of the reflecting region of the ionosphere. The results of a number of works (see, for example, \(^{5,6,15}\)) show, however, that the basic assumption expressed by formula (1) agrees rather well with experiment and, thus, may be taken as the basis for the further analysis of the measurement results.

a) The degree of turbidity of the ionosphere \(\alpha\)

The parameter \(\beta^2\) introduced above, as we have seen, is equal to the ratio of the energy of the specularly reflected wave to the energy of the scattered signals, i.e., it characterizes the measure of inhomogeneity of the ionosphere, or, one may say, the degree of its turbidity. Here, however, we shall agree to characterize the degree of turbidity of the ionosphere not by the coefficient \(\beta^2\), but by means of the coefficient

\[ \alpha=\frac{\sum a_s^2}{\sum a_s^2+a_0^2}=\frac{1}{1+\beta^2}, \tag{11} \]

i.e., by the ratio of the energy of the scattered waves to the total energy of the “reflected” wave, equal to the sum of the energy of the scattered waves and the energy of the specularly reflected wave.

In one of the works \(^{4}\) it was found that in approximately half the cases of observations of reflection from different layers \(\beta^2\) varied within the limits from 0 to 1, which corresponds to a variation of the coefficient of turbidity

\[ \alpha \simeq 1 \div 0.5. \tag{12} \]

In other experiments \(^{15}\), very few in number, values were obtained predominantly

\[ \beta^2 \simeq 1.3 \div 3, \]

i.e.

\[ \alpha \simeq 0.3 \div 0.1. \tag{13} \]

Further, as a result of a large number of measurements \(^{6}\), it was established that in the \(F_2\) layer under different conditions \(\beta^2 \simeq 0 \div 50\), i.e.,

\[ \alpha \simeq 1 \div 0.02. \tag{14} \]

At present, the published measurement results do not yet make it possible to establish what value of \(\alpha\) occurs most often in different layers of the ionosphere and how \(\alpha\) changes depending on the time of day, season, etc.

b) The root-mean-square velocity \(v_0\) of chaotic motions in the ionosphere

Since the results of experiments show that the structure of the reflecting region is such that the assumption (1) made above is correct, one can therefore make various assumptions concerning the cause of the frequency shift \(f_0\) of the incident wave upon scattering.

Ya. L. Alpert

Suppose that the scattering centers move chaotically and that the frequency shift is the result of the Doppler effect, so that in the case of vertical incidence on the ionosphere

\[ f_s=f_0\pm \frac{2v_s}{\lambda}, \tag{15} \]

where \(v_s\) is the normal component of the velocity of motion of the \(s\)-th scattering center. The chaotic nature of the motion means that the velocities of the scattering centers have a Gaussian distribution, i.e.,

\[ W(v)=\frac{1}{\sqrt{2\pi}\cdot v_0}\,e^{-\frac{v^2}{2v_0^2}}, \tag{16} \]

where

\[ v_0^2=\overline{v^2}. \tag{17} \]

Proceeding from this, one can now write that the total energy of the scattered waves is

\[ W_0=\frac{1}{2}\sum a_s^2=\int_0^\infty W(f)\,df \tag{18} \]

and that their energy spectrum is

\[ W(f)=\frac{W_0}{\sqrt{2\pi}\,\sigma_f}\, e^{-\frac{(f-f_0)^2}{2\sigma_f^2}}, \tag{19} \]

where

\[ \sigma_f^2=\overline{(f-f_0)^2}=\frac{4v_0^2}{\lambda^2}. \tag{20} \]

It seems important to derive a formula that makes it possible to determine, from the results of measurements of the amplitude of the unit signal, the value \(v_0\). This can be done, in particular, by calculating successively the value \(\overline{\left(\dfrac{dR^2}{dt}\right)^3}\). Omitting here the corresponding calculations, which were carried out in the cited work\(^6\), one can obtain that, to within a factor of order \(\sqrt{\dfrac{\pi}{2}}\),

\[ v_0= \frac{\lambda\cdot \overline{R}\cdot |\Delta R|_\tau} {4\cdot \pi\cdot \tau \sqrt{\dfrac{\pi}{2W_0^2+a_0^2W_0}}}, \tag{21} \]

where

\[ |\Delta R|_\tau=\overline{|R(t)-R(t+\tau)|} \]

is the mean value of the modulus of the difference of successive values of the amplitude \(R_n\) and \(R_{n+1}\), measured at some instant \(t\) and at the following instant \((t+\tau)\).

In the case when \(a_0=0\), i.e., when the influence of the specularly reflected wave may be neglected, taking into account that in this case

\[ W_0=\frac{1}{2}\sum a_s^2=\frac{1}{2}\overline{R^2}=\frac{2}{\pi}(\overline{R})^2, \]

we obtain from (21):

\[ v_0=\frac{\lambda\,\overline{|\Delta R|}}{8\sqrt{2}\,\tau\,\overline{R_i}}. \tag{22} \]

Values of the root-mean-square velocity \(v_0\) of chaotic motions, calculated for \(a_0\ne0\) and \(a_0=0\), have been given in the literature. In work\(^6\), in processing the measurement results by means of formula (21), values of \(v_0\) were obtained for the \(F\) layer varying within the limits

\[ v_0 \cong (0.2 \div 15)\ \text{m/sec}, \tag{23} \]

and the values of \(v_0\) encountered most often in these experiments apparently lay within the limits \((1 \div 4)\ \text{m/sec}\).

In another work\(^5\), by means of formula (22), in which the factor \(\frac{1}{\sqrt{2}}\) was absent*), the values

\[ v_0 \cong (2 \div 6)\ \text{m/sec}, \tag{24} \]

were calculated, while the mean value of \(v_0\) proved to be equal to \((2 \div 3)\ \text{m/sec}\). By means of formula (22), from the data of other experiments\(^4\), values were also obtained

\[ v_0 \cong (0.5 \div 5)\ \text{m/sec} \tag{25} \]

with mean value \(v_0 \cong 2\ \text{m/sec}\), and the values of \(v_0\) encountered most often were of the order of \((1 \div 2)\ \text{m/sec}\).

At present there are still no data that would characterize more precisely the variability of \(v_0\) in different layers of the ionosphere or its dependence on various conditions. There are only some indications that at local noon \(v_0\) decreases\(^6\).

§ 3. STATISTICAL INVESTIGATIONS OF THE AMPLITUDE AND PHASE OF SIGNALS BY MEANS OF THE CORRELATION COEFFICIENT

In the preceding paragraph a statistical method was considered for processing the results of measurements of the amplitude of signals reflected from the ionosphere (see\(^6\)), which makes it possible to calculate from these data the degree of turbidity \(\alpha\), characterizing the degree of inhomogeneity of the “quiet” (see below § 4) ionosphere, and the root-mean-square velocity \(v_0\) of the chaotic motions occurring in it, characterizing the mobility and variability of the inhomogeneities present in it.

* For a reason unclear to the author.

Let us now consider another statistical method for analyzing experimental data, which in principle makes it possible to compute not only the parameters indicated above, but, with an appropriate arrangement of experiments, also a number of other quantities whose knowledge is important for studying the structure of the ionosphere. This is the correlation coefficient (see 15, 16), or, in other words, the autocorrelation function of the amplitude or phase-difference of unit signals. The qualification made above, “in principle,” is connected with the fact that at present this method of calculation does not yet have sufficient generality, since only the case has been consistently brought to convenient computational formulas in which the distribution of the field intensity is Gaussian (and correspondingly the distribution of signal amplitudes is Rayleigh), i.e. when there is no specularly reflected wave \((a_0=0)\) of constant frequency \(\omega_0\), having some constant fixed direction of arrival at the point of observation. In addition, the corresponding calculations have not been carried out (as in § 2) for the case in which there are simultaneously chaotically moving scattering centers and a steady drift of the reflecting region. Therefore at present this method is less general than the one described in § 2. The data obtained with its aid are often not quite accurate, since the experiments, as we saw in § 2, show that for the most part \(\beta^2 \sim (1 \div 2)\) and that the component of the field \(a_0 \cos \omega_0 t\) can by no means be neglected.

It is well known that the correlation coefficient of two sequences of numbers \(R_1, R_2\), which in our case may be, for example, the values of the amplitude at the instant \(t\) \((R_1=R(t))\) and at the instant \((t+\tau)\) \((R_2=R(t+\tau))\), is equal to

\[ \rho_R(\tau)= \frac{\overline{R(t)R(t+\tau)}-(\overline{R(t)})^2} {\overline{R(t)^2}-(\overline{R(t)})^2} = 1-\frac{\overline{(\Delta R)^2_\tau}} {2\{\overline{R(t)^2}-(\overline{R(t)})^2\}}. \tag{26} \]

Sometimes \(\rho_R(\tau)\) is written in the form

\[ \rho_R(\tau)= \frac{\overline{[R(t)-\overline{R(t)}]\,[R(t+\tau)-\overline{R(t)}]}} {[R(t)-\overline{R(t)}]^2}, \tag{27} \]

or, in a more general form,

\[ \rho_R(\tau)= \frac{ \displaystyle \int_{-\infty}^{\infty} R(t)R(t+\tau)\,dt -\left\{\displaystyle \int_{-\infty}^{\infty} R(t)\,dt\right\}^2 }{ \displaystyle \int_{-\infty}^{\infty} R(t)^2\,dt -\left\{\displaystyle \int_{-\infty}^{\infty} R(t)\,dt\right\}^2 }. \tag{28} \]

For the Rayleigh distribution of \(R(t)\), considered, as was already indicated above in this section, i.e., for the case when

\[ W(R)=\frac{2R}{\overline{R^2}}e^{-\frac{R^2}{\overline{R^2}}} \tag{29} \]

and

\[ \overline{R^2}=\frac{4}{\pi}\left(\overline{R}\right)^2=\sum a_s^2, \tag{30} \]

the distribution of the difference

\[ \{R(t)-R(t+\tau)\}=(\Delta R)_\tau \]

must be Gaussian, i.e.,

\[ \overline{(\Delta R_\tau)^2}=\frac{\pi}{2}\left(\overline{|\Delta R_\tau|}\right)^2. \tag{31} \]

Therefore formula (26) takes the form

\[ \rho_R(\tau)=1-\frac{\pi^3\left(\overline{|\Delta R|_\tau}\right)^2}{4(4-\pi)\left(\overline{R}\right)^2}. \tag{32} \]

We see that, having experimental curves of the type shown above in Fig. 3, it is easy to construct the curve of the dependence of the correlation coefficient on \(\tau\). Naturally, one can also calculate the correlation coefficient between two series of values of the amplitudes of signals measured at one and the same time, but at different observation points mutually separated, for example, by a distance \(\xi\), i.e., in the one-dimensional case at the points \(x\) and \((x+\xi)\). The correlation coefficient is then written in the form \(\rho_R(\xi)\). In what follows we shall distinguish these two correlation coefficients, calling \(\rho_R(\tau)\) the temporal, and \(\rho_R(\xi)\) the spatial correlation coefficients.

Let us now consider how the values of the correlation coefficient, calculated from experimental data, can be used to determine the parameters of the reflecting region of the ionosphere that are of interest to us.

a) Temporal correlation coefficient \(\rho_R(\tau)\) and the velocity \(v_0\) of chaotic motions

The field strength \(E(t)\), written above with the aid of formula (1), can be represented in the form

\[ E(t)=R(t)\cos\{2\pi f_0 t+\psi(t)\}, \tag{33} \]

i.e., as a quasi-periodic oscillation with amplitude and phase (or frequency) slowly varying in time. The term slowly, as is known, is understood in the sense of slowness of change with respect to

compared with the carrier frequency $f_0$. The fact that the “period” of variation of the amplitude of the oscillations is large in comparison with the wave period $T_0=\dfrac{1}{f_0}$ is clearly seen, in particular, from Fig. 3. The slowness of the variation of $\psi(t)$ is manifested experimentally in the fact that the rms velocity $v_0$ is small, so that the rms frequency displacement is

\[ \Delta f_0=\sqrt{\overline{(f-f_0)^3}}=\frac{2v_0}{\lambda}\ll f_0 . \tag{34} \]

Further, since for a chaotic distribution of velocities $v_s$ all directions of the velocity vector are equiprobable, i.e. the directions of motion of the scattering centers toward the observer and away from the observer are equiprobable, the values $\Delta f_0>0$ and the values $\Delta f_0<0$ are therefore equiprobable. On the basis of what has been said, it may be asserted that the energy spectrum $W(f)$ of the quasiperiodic oscillation (33) is a narrow function symmetric with respect to $f_0$.

For this type of energy spectrum of chaotic oscillations, taking into account that the distribution of oscillation amplitudes is Rayleighian ($a_0=0$), i.e. is described by (29), it has been shown\(^{16}\) that, with a sufficient degree of accuracy,

\[ \rho_R(\tau)\cong \frac{ \left\{ \displaystyle\int_{-\infty}^{\infty} W(f+f_0)e^{i2\pi f\tau}\cdot df \right\}^{2} }{ \left\{ \displaystyle\int_{-\infty}^{\infty} W(f)\cdot df \right\}^{2} }. \tag{35} \]

If we now use formula (19) for $W(f)$, then in this case we obtain

\[ \rho_R(\tau)=e^{-\frac{16\pi^{2}\tau^{2}v_0^{2}}{\lambda^{2}}}. \tag{36} \]

Formula (36) contains one unknown parameter, namely the value of the rms velocity $v_0$, which is easily calculated by comparing (36) with (32) or (26), which determine from the experimental data the value of the correlation coefficient $\rho_R(\tau)$. If the time interval $\tau$ over which the amplitude values are counted off is such that the correlation coefficient is close to unity, then, using only the linear term in the expansion of the right-hand side of (36) in a series, one can obtain, by combining (36) with (32), the following simple formula determining $v_0$:

\[ v_0=\frac{\lambda\cdot |\Delta R|_{\tau}}{8\cdot \sqrt{4-\pi\bar{R}}}. \tag{37} \]

Comparison of (37) with formula (22), derived in § 2,б), shows that they differ from one another by a numerical factor of order 1.5.

The method described in this paragraph for processing experimental data was used in the works cited above \(^{4}\) and \(^{5}\) to determine the values \(v_0\) *), given in the preceding paragraph.

When analyzing experimental data, it is naturally also of interest to compute \(v_0\) from measurements of the amplitude of the twice-reflected wave. This can serve not only as a check on the correctness of the calculations, but also as a criterion determining how strongly scattering near the earth’s surface affects the twice-reflected wave. In the case of insignificant scattering near the earth’s surface, identical values of \(v_0\) should be obtained from the measurement data for the amplitudes of both signals. Calculations using theorem (35) lead to a very simple formula establishing the relation between the correlation coefficient \(\rho_{R_1}(\tau)\) of the once-reflected wave and the correlation coefficient \(\rho_{R_2}(\tau)\) of the twice-reflected wave. It is obtained \(^{16}\) that

\[ \rho_{R_2}(\tau)=\{\rho_{R_1}(\tau)\}^2 . \tag{38} \]

б) Spatial correlation coefficient and angular spread \(\theta_0\) of the beam of scattered waves

Let us now examine what is obtained when processing measurement results with the aid of two separated antennas, when the correlation coefficient is computed from the experimental data

\[ \rho_R(\xi)=1-\frac{\pi^3\left(\left|\overline{R'(x)-R'(x+\xi)}\right|\right)^3}{4(4-\pi)(\overline{R})^2}. \tag{39} \]

In this case it is useful to approach the analysis of the experimental data somewhat differently. It was already indicated above that, owing to the inhomogeneity of the reflecting region, not one reflected wave arrives at the observation point, but a beam—a cone of waves, each component of which has its own direction, i.e. its own value of the angle \(\alpha\), for example, with the line joining the separated observation points. If we restrict ourselves to considering the linear and coplanar case, i.e. consider a beam of waves arriving only in one plane, then we may write that the total energy of the beam of “reflected” waves is equal to

\[ W_0=\int_0^{2\pi} W(\alpha)\,d\alpha, \tag{40} \]

*) It is not clear why, in these works, in the formula analogous to formula (37), the factor \(\sqrt{4-\pi}\) is absent.

where \(W(a)\) is the function determining the angular distribution of the beam of waves.

In this connection one should keep in mind two types of experiments. In one of them the angular distribution of the beam of waves is investigated in a vertical plane, when it is grouped about the vertical direction or an inclined direction. In this case the spaced antennas lie in the same plane—this is an experiment for determining the inclination of the front of the arriving wave (see Fig. 6, a). In another experiment the angular distribution of the beam of waves is investigated in an inclined or horizontal plane (if the elevation angle \(\vartheta\) approaches zero), containing the direction of the front of each wave and perpendicular to the vertical plane. In this case the spaced antennas are arranged so that the line joining them is perpendicular to the vertical plane (see Fig. 6, b). This is a direction-finding experiment—for determining the scatter of the azimuth of the arriving wave. In both experiments one can measure not only the amplitude of the signal at the spaced points, but also the phase difference of the oscillations received at them.

Fig. 6. \(a\)—scheme of an experiment for determining the scatter of the angles \(\theta_0\) in a vertical plane, when the inclination of the wave front is measured; \(b\)—scheme of a direction-finding experiment, when the scatter of the angles \(\theta_0\) in inclined or horizontal planes is determined.

For a given angular distribution \(W(s)\), where \(s=\sin\alpha\), it is proved in \(^{16}\) that for the Rayleigh distribution of amplitudes we have, with a high degree of accuracy:

\[ \rho_R(\xi)= \frac{ \left\{ \int_{-\infty}^{\infty} W(s)e^{i2\pi\cdot s\frac{\xi}{\lambda}}\,ds \right\}^{2} }{ \left\{ \int_{-\infty}^{\infty} W(s)\,ds \right\}^{2} }. \tag{41} \]

The use of this theorem makes it possible to analyze the results of measurements in more detail and to calculate a number of parameters characterizing the structure of the ionosphere. In this paragraph only one such possibility will be considered; other possibilities will be analyzed in § 4, b), c).

Suppose that the beam of rays has a narrow symmetric distribution relative to some mean direction \(\alpha_0\), i.e., that

\[ \alpha=\alpha_0+\theta \tag{42} \]

and

\[ W(\theta)=W(-\theta), \tag{43} \]

where \(\theta\) is a small quantity \((\sin\theta\sim\theta)\), and the angle \(\alpha_0\sim \dfrac{\pi}{2}\)—in the experiment for determining the inclination of the wave front under vertical reflection from the ionosphere and in the pleiades experiment, when the spaced antennas are set perpendicular to the direction of arrival of the wave at the observation point. This case, apparently, should occur when the dimensions of the scattering centers are sufficiently large in comparison with the wavelength. The measurement data given below show that this assumption usually corresponds to real conditions.

The use of (42) and (43) reduces equation (41) to a simpler form; namely, one obtains \(^{15}\)

\[ \rho_R(\xi)= \frac{ \left\{ \displaystyle\int_{-\infty}^{\infty} W(\theta)\cdot e^{\,i2\pi\theta\sin\alpha_0\frac{\xi}{\lambda}}\,d\theta \right\}^{2} }{ \left\{ \displaystyle\int_{-\infty}^{\infty} W(\theta)\,d\theta \right\}^{2} }. \tag{44} \]

Now assuming, analogously to the way this was done above for the energy partial spectrum, that the angular distribution of the beam of waves is Gaussian, i.e., that

\[ W(\theta)=\frac{W_0}{\sqrt{2\pi}\,\theta_0}\, e^{-\frac{\theta^{2}}{2\theta_0^{2}}}, \tag{45} \]

where

\[ \theta^{2}=\theta_0^{2}, \tag{46} \]

we obtain from (44):

\[ \rho_R(\xi)=e^{-\frac{4\pi^{2}\xi^{2}\theta_0^{2}\sin^{2}\alpha_0}{\lambda^{2}}}. \tag{47} \]

Comparison of (47) with (39) thus makes it possible to determine, from the results of measurements of the signal amplitude, the value of \(\theta_0\), i.e., the root-mean-square deviation of the angles of the beam of waves from the mean direction or, in other words, the angular spread of the beam of scattered waves. If \(\rho_R(\xi)\sim 1\), which depends on the distance \(\xi\) chosen in the experiments, then a simple formula is obtained that directly determines \(\theta_0\), namely

\[ \theta_0=\frac{\lambda\cdot |\Delta R|_\xi}{4\sqrt{\,4-\pi\cdot \xi\cdot R\sin\alpha_0\,}}, \tag{48} \]

where

\[ |\Delta R|_\xi=\left|R(x)-R(x+\xi)\right|. \]

Similar calculations can also be carried out for the case in which the phase difference of the \(\psi\)-oscillations received at two separated points is measured. For this case it is obtained\(^{15}\) that

\[ |\psi|=\arccos\sqrt{\rho_R(\xi)}, \tag{49} \]

or, using (47) and when \(\rho_R(\xi)\sim 1\),

\[ \left(|\psi|\right)^2\sim 2\left(1-\sqrt{\rho_R(\xi)}\right) \tag{50} \]

and

\[ \theta_0=\frac{\lambda\cdot |\psi|}{2\pi\cdot \xi\sin\alpha_0}. \tag{51} \]

It should be pointed out that, when conducting direction-finding-type experiments, where \(\theta_0\) is determined in the inclined plane (see above), the value of the angular spread in the horizontal plane is, as is not difficult to see, equal to

\[ \theta_{0,\mathrm{h}}\simeq \frac{\theta_0}{\cos\vartheta}, \tag{52} \]

where \(\vartheta\) is the elevation angle of the inclined plane.

By the methods described here, the results of a few-point set of experiments carried out in the range \((4\div 7)\) MHz were processed, when reflections from the \(F\) and \(E_{\mathrm{spor}}\) layers\(^{16}\) were observed. The angles of incidence in the vertical plane were measured when the reflection was almost vertical. The values obtained were

\[ \theta_0\sim (0.5\div 3)^\circ. \tag{53} \]

In another work\(^{17}\), more systematic observations were carried out at fixed frequencies of 2.4 and 4.8 MHz. The results of measurements upon reflection from different layers were processed separately. The processing method for the experimental results was based on formulas (40) and (41); however, for \(W(s)\) the value \(\cos^n\vartheta\) was adopted, and not formula (45). Some recalculation of the data available in that work shows that analogous values, in order of magnitude, are obtained if one uses formula (45) and the resulting

from it by formula (48). The results of these experiments are shown in Fig. 7, which gives the distribution curves of the measured values of \(\theta_0\) for different cases. From these data it is evident that the most frequently occurring value of \(\theta_0\) for reflections from the \(E\) and \(F\) layers at a frequency of \(2.4\) Mc/s is

\[ \theta_0 \simeq 5^\circ, \tag{54} \]

and at a frequency of \(4.8\) Mc/s, for reflection from the \(F\) layer,

\[ \theta_0 \simeq 2.5^\circ. \tag{55} \]

As we see, rather few results of measurements of \(\theta_0\) have been published, but they already make it possible to draw certain conclusions, which are indicated below.

§ 4. STRUCTURE OF THE REFLECTING REGION OF A “COMPLETELY UNDISTURBED” IONOSPHERE

It has already been pointed out above that, in order to draw correct conclusions from the results of measurements, it is necessary to investigate the individual signals reflected from the ionosphere, i.e., one magnetically split component of the wave. However, this condition is still insufficient. Very often such an individual signal in fact represents a group of signals creeping over one another. Sometimes it can be seen that this group contains two or three, and in some cases an even larger number of signals (see below, § 4, d)). It is clear that such phenomena, even in an incipient state, i.e., when the group of signals is still small—consisting of only two or three signals—testify to a more complicated structure of the reflecting layer and to a complicated redistribution of the energy of the wave incident on the ionosphere.

Within the scope of the questions considered here, we shall proceed from the assumption that such a complication of the signal structure is the result of the ionospheric state becoming “disturbed.” We shall consider the ionosphere “quiet,” or “completely undisturbed,” in those cases when the individual reflected signal is not split at all, which ensures the correct interpretation of the experimental results on the basis of the theoretical formulas given above. Naturally,

Fig. 7. Distribution of the results of measurements of the angular spread \(\theta_0\) at a frequency of \(2.4\) Mc/s upon reflection:
\(a\)—from the \(E_{\mathrm{spor}}\) layer at night,
\(b\)—from the \(E\) layer by day,
\(v\)—from the \(F\) layer at night,
and \(g\)—at a frequency of \(4.8\) Mc/s upon reflection from the \(F\) layer by day and by night.

that the term “quiet” ionosphere is rather conventional, if only because, in the ionosphere, as we have seen, there continuously occur motions of scattering centers, replacement of some inhomogeneities by others, and similar phenomena. Here we shall be concerned mainly with the structure of the reflecting region of the quiet ionosphere and, consequently, with those conclusions that can be drawn from an analysis of the results of observations of a single signal. It should be noted that the method of recording reflected signals used in works \(^{4,5}\), unlike the method of work \(^{6}\), does not always guarantee that the measurement results under consideration correspond to a quiet ionosphere in the sense in which we have defined it above.

Proceeding from what has been said, we shall henceforth consider that the reflecting region is that part of the ionosphere which mainly participates in the formation of the single signal obtained as a result of “reflection” from the layer, i.e., the region which returns to the point of observation the greatest part of the energy of the wave incident on the layer.

Let us consider how one can determine the dimensions and investigate the internal structure—or, in other words, the fine structure—of such a reflecting region, and what conclusions can be drawn from the corresponding analysis of the experimental data given in the works cited above.

a) Dimensions of the reflecting region

If the reflecting layer is smooth, not rough, then the size of the region that forms the vertically reflected signal is naturally equal to the radius of the first Fresnel zone, i.e.,

\[ r_0 \sim \sqrt{\lambda z_0}, \tag{56} \]

and the thickness of this region \(\Delta z_0\) is, in order of magnitude, equal to the wavelength \(\lambda\) in the ionosphere. Here \(z_0\) corresponds to the height at which the refractive index \(n\) for the given wavelength satisfies the condition \(n \sim 0\) and where total reflection of the wave occurs. In this case the reflected signal is not broadened, if we exclude from consideration the spreading of the signal caused by the dispersion of the ionosphere. However, these values \(r_0\) and \(\Delta z_0\) are the smallest possible. Under real conditions the reflecting region is rough. Therefore, at the point of observation the “reflected” signal is formed as a result of the summation of many “elementary” waves scattered by the inhomogeneities contained in it, which may also come from parts of the layer lying beyond the limits of the first Fresnel zone. The dimensions of this region, in particular, depend on the ratio of the sizes of these inhomogeneities to the wavelength, which determines the character of the directivity pattern—

STATISTICAL CHARACTER OF THE STRUCTURE OF THE IONOSPHERE

…of the secondary radiation of the scattering center. Naturally, in this case the “reflected” unit signal must broaden, and the magnitude of its broadening can be used to estimate the dimensions of the reflecting region.

Indeed, suppose that the “width” of the incident pulse is equal to \(\tau_0\), and the width of the “reflected” pulse is equal to \(\tau'\); then the thickness of the region that can cause this signal broadening is

\[ \Delta z_0 = z_0 - z_{\mathrm{n}}, \tag{57} \]

where \(z_0\) is its upper boundary—the height at which the refractive index of the wave \(n = 0\), and \(z_{\mathrm{n}}\) is its lower boundary (Fig. 8). Thus,

Fig. 8. Diagram illustrating the method of calculating the thickness \(\Delta z_0\) and the radius \(r_0\) of the reflecting region of the ionosphere.

Fig. 9. Diagram of the so-called parabolic layer.

\(z_0\) and \(z_{\mathrm{n}}\) are respectively the upper and lower limits of the integral determining the time \(\tau' - \tau_0\), namely

\[ \Delta \tau_0 = \tau' - \tau_0 = \int_{z_{\mathrm{n}}}^{z_0} \frac{dz}{u(z,f)}, \tag{58} \]

where \(u\) is the group velocity of the incident wave having frequency \(f\).

Let us calculate \(\Delta z_0 = z_0 - z_{\mathrm{n}}\) for a layer of parabolic form, which, as is known, often approximates the form of real ionospheric layers quite well and for which

\[ n^2 = 1 - M^2 \left\{ 1 - \left( \frac{z_m - z}{z_m} \right)^2 \right\}^3, \tag{59} \]

where

\[ M = \frac{f_c}{f} \tag{60} \]

is the ratio of the critical frequency of the layer \(f_c\) to the working frequency \(f\), and \(z_m\) is the semi-thickness of the layer (Fig. 9).

Substituting into (58) the known relation*)

\[ u=c\cdot n \tag{61} \]

and taking into account that

\[ z_0=z_m\left(1-\frac{\sqrt{M^2-1}}{M}\right), \tag{62} \]

(this follows from (59) for \(n=0\)), we find from (58)

\[ \Delta\tau_0=\frac{z_m}{cM}\ln\left\{ \frac{\dfrac{z_m-z_{\mathrm n}}{z_m}+ \sqrt{1-M^2+M\left(\dfrac{z_m-z_{\mathrm n}}{z_m}\right)^2}} {\sqrt{M^2-1}} \right\} \tag{63} \]

and

\[ \Delta z_0=z_0-z_{\mathrm n} =z_m\left(1-\frac{\sqrt{M^2-1}}{M} -\frac{1-(M-K)^2}{2KM}\right), \tag{64} \]

where

\[ K=\exp\left[\frac{(c\Delta\tau_0)M}{z_m}+\ln\sqrt{M^2-1}\right]. \tag{65} \]

Since the value of \(\Delta z_0 \ll z_0\), it may be approximately assumed that

\[ r_0\simeq \sqrt{r_1^2-z_0^2} \simeq \sqrt{(z_0+\Delta z_0)^2-z_0^2} \simeq \sqrt{2z_0\Delta z_0} \tag{66} \]

(see Fig. 8). This formula naturally gives values of \(r_0\) correct in order of magnitude if \(r_0 \ll z_0\).

From formulas (64) and (65) it is seen that the radius \(r_0\) of the reflecting region depends on \(z_m\), \(f_c\), and \(\Delta\tau_0\).

Analysis of the results of a number of measurements shows that, under conditions of a quiet ionosphere, the value

\[ c\Delta\tau_0 < 0.2\cdot c\tau_0, \tag{67} \]

i.e. the signal is broadened by no more than 20%. Since the width of the emitted pulse in the measurements described above was of the order of \(70\cdot 10^{-6}\) sec, \(c\Delta\tau_0\sim 4\) km.

Table I

Values of the radius \(r_0\) of the reflecting region of the ionosphere (in km) for various \(M\)

	1.05	1.1	1.2	1.5
Layer \(E\)	5.2	6.3	8.0	11
Layer \(F\)	2.7	3.1	4.3	5.7

Table I gives the results of calculating \(r_0\) by means of formulas (66) and (64) for layer \(E\) (\(z_m=20\) km) and layer \(F\) (\(z_m=200\) km) for \(c\Delta\tau_0=4\) km.

It is seen from the table that, as the maximum of the layer is approached (\(M\) decreases), \(r_0\) decreases. Moreover, owing to the small semi-thickness of layer \(E\), it follows that the values of \(r_0\) are larger for layer \(E\) than for layer \(F\).

*) We neglect here the influence of the Earth’s magnetic field.

(in the calculations it was assumed that the height of the beginning of the layer \(E\) is equal to \(110\) km and that of the layer \(F\) to \(250\) km). The inverse result is obtained in calculating the radius of the Fresnel zone with the aid of formula (56).

Naturally, the calculation given for the values of \(r_0\) is approximate in character and becomes quite incorrect when \(c\Delta\tau_0\) decreases, since then \(\Delta z_0 \to 0\) and \(r_0 \to 0\). It is obvious that the estimates made are valid only when the values of \(r_0\) are greater than, or of the order of, the radius of the first Fresnel zone.

Let us now see what values of \(r_0\) are obtained from the results of measuring the values of the angular spread \(\theta_0\) given above. From these data it is seen that in one of the experiments values \(\theta_0 \sim (0.5 \div 3)^\circ\) were obtained (see (53)), whence it follows that \(r_0\) varied under these conditions, for different layers, within the limits

\[ r_0 \simeq z_0\theta_0 \sim (1 \div 10)\ \text{km}. \tag{68} \]

In other experiments it was found (see (54), (55)) that the most frequently occurring values of \(\theta_0\), respectively for the layers \(E\) and \(F\), were equal to \(5^\circ\) and \(2.5^\circ\), i.e.

\[ r_0 \simeq (8 \div 12)\ \text{km}. \tag{69} \]

Thus from these data, which are still very preliminary, it is seen that, in order of magnitude, they are in general agreement with the values of \(r_0\) calculated in Table I.

b) Dimensions of the Scattering Irregularities and the Fine Structure of the Ionosphere

It follows from the preceding that, under conditions of an undisturbed ionosphere, the radius of the reflecting region of the ionosphere apparently has the mean value

\[ r_0 \sim 5\text{--}6\ \text{km}. \tag{70} \]

However, we have seen that the change in amplitude of the signal reflected from the ionosphere indicates that within one such region chaotic motions of scattering centers take place, i.e. that the reflecting region has a fine structure and, apparently, consists of inhomogeneities whose dimensions are considerably smaller than \(r_0\). Let us see what conclusions can at present be drawn about the mean or most probable dimensions of these inhomogeneities.

The very fact that the behavior of the field of the reflected wave has a statistical character and is well described by equation (1) indicates that the reflecting region contains a sufficiently large number of scattering centers, namely such a number as ensures fulfillment of the law of large numbers. Suppose

therefore, since the scattering centers do not interact with one another, one may assume that the reflecting region must consist of at least 200–300 alternating condensations and rarefactions of the ionization density. On this basis, we obtain, according to (70), that the radius of the scattering inhomogeneity may have the value

\[ \xi_0 \sim (300\text{—}400)\ \text{m}. \tag{71} \]

This estimate is, of course, roughly qualitative; however, as we shall see, the value (71) agrees quite well with more exact determinations of \(\xi_0\). Let us consider these data.

Above, formula (47) was given for the correlation coefficient \(\rho_R(\xi)\) of the amplitude of signals received at two separated points located on the earth’s surface. This expression may be used to calculate the value of \(\xi_0\), on the basis of the following considerations.

The diffraction pattern observed at the earth’s surface may be regarded as the result of diffraction from a rough screen upon which a beam of parallel rays is incident, i.e. as Fresnel-type diffraction. Such conditions actually occur, since the distance \(z_0\) to the diffracting screen (the ionospheric layer) amounts to many wavelengths (several hundreds and more), and the linear dimensions of the reflecting region \(r_0\) are also much smaller than \(z_0\). For this case it is proved\(^{16}\) that, if the linear dimensions of the inhomogeneities are greater than the wavelength, then the correlation coefficient has the same values both in the immediate vicinity of the diffracting screen and on it itself, and also in a plane removed from the screen by any distance. For the case in which the dimensions of the inhomogeneities are less than the wavelength, i.e. when the radiation of the diffracting screen is isotropic, this theorem is valid only beginning with distances removed from the screen by several wavelengths. In the case considered here the condition \(2\xi_i \gg \lambda\) is satisfied; therefore, to a high degree of accuracy it may be assumed that the correlation coefficient measured at the earth’s surface is equal to the correlation coefficient in the reflecting region of the ionosphere.

We shall now take the size of an inhomogeneity to correspond to that distance \(\xi=\xi_0\) at which

\[ \rho_R(\xi_0) \cong e^{-1}. \tag{72} \]

In work\(^{17}\) \(\xi_0\) is defined as the distance at which \(\rho_R(\xi_0) \cong 0.5\). Using formula (47), we thus obtain that

\[ \xi_0 \cong \frac{\lambda}{2 \cdot \theta_0 \sin \alpha_0}. \tag{73} \]

Now, substituting into (73) the values of \(\theta_0\) given above (see (53)), we have:

\[ \xi_0 \simeq (200 \div 1000)\ \text{m}. \tag{74} \]

The use of the results of the measurements of work \(^{17}\) (see (54) and (55)) leads to the value

\[ \xi_0 \simeq 380\ \text{m}. \tag{75} \]

In work \(^{17}\), as has already been indicated, \(\xi_0\) was determined as the distance at which \(\rho_R(\xi) \sim 0.5\); therefore the values of \(\xi_0\) calculated there are smaller than the value (75). Figure 10 gives the distribution curves of the results of determining the linear dimensions \(\xi_0\) of inhomogeneities obtained in that work. It is evident from the figure that the most frequently occurring value of \(\xi_0\) is of the order of 200 m.

c) Lifetime of “inhomogeneities” in the ionosphere

It has already been assumed earlier that, within the reflecting region of the ionosphere, some inhomogeneities must be continuously replaced by others, and that the time during which inhomogeneities of small size are dissipated is considerably less than the duration of the experiment. We shall now calculate what lifetimes of inhomogeneities should be expected in the different layers of the ionosphere. To carry out this calculation, let us assume that at the initial moment \(t_0=0\) the inhomogeneity has a spherical form of radius \(R_0\), and that its electron concentration is equal to \(kN_0\), where \(k>1\) and \(N_0\) is the electron concentration of the surrounding medium. It is easy to show that the fastest process leading to the dissipation of this inhomogeneity will be diffusion; therefore, for calculating the lifetime of a spherical inhomogeneity one may use the solution of the diffusion equation

\[ \frac{\partial N}{\partial t}=D\cdot\nabla^2 N(R,t), \tag{76} \]

Fig. 10. Distribution of the results of measuring the linear dimensions of inhomogeneities \(\xi_0\) at a frequency of 2.4 MHz upon reflection:

a—from the sporadic \(E\) layer at night, b—from the \(E\) layer by day, c—from the \(F\) layer at night, and d—at a frequency of 4.8 MHz upon reflection from the \(F\) layer by day and night.

which has the form

\[ N(R,t)=\frac{1}{R\sqrt{\pi}}\int_{-\infty}^{\infty} N(R+2\eta\sqrt{Dt})\,(R+2\eta\sqrt{Dt})e^{-\eta^2}\,d\eta, \tag{77} \]

where \(D\) is the diffusion coefficient, \(R\) is the distance from the center of the sphere, and \(\eta\) is the variable of integration.

Since we have assumed that at the initial moment \(t_0=0\)

\[ \begin{aligned} &N(R,0)=N_0 \quad &&\text{for } R\geq R_0\\ &\text{and}\qquad N(R,0)=kN_0 \quad &&\text{for } R\leq R_0 \end{aligned} \left\} \tag{78} \right. \]

then from (77) we obtain:

\[ N(R,t)=\frac{N_0}{R\sqrt{\pi}} \left\{ \left[ \int_{-\infty}^{\frac{R_0+R}{2\sqrt{Dt}}} (R+2\eta\sqrt{Dt})e^{-\eta^2}\,d\eta+ \right.\right. \]

\[ \left.\left. +k\int_{-\frac{R}{2\sqrt{Dt}}}^{-\frac{R_0+R}{2\sqrt{Dt}}} (R+2\eta\sqrt{Dt})e^{-\eta^2}\,d\eta+ \int_{\frac{R_0-R}{2\sqrt{Dt}}}^{\infty} (R+2\eta\sqrt{Dt})e^{-\eta^2}\,d\eta \right] \right\} \tag{79} \]

or

\[ N(R,t)=\frac{1}{2} \left\{ 2+(k-1)\left[ \psi\left(\frac{R_0-R}{2\sqrt{Dt}}\right) +\psi\left(\frac{R_0+R}{2\sqrt{Dt}}\right) \right] -\frac{\sqrt{Dt}}{r\sqrt{\pi}}(k-1) \left[ e^{-\frac{(R_0-R)^2}{4Dt}}- e^{-\frac{(R_0+R)^2}{4Dt}} \right] \right\}, \tag{80} \]

where

\[ \psi(a)=\frac{2}{\sqrt{\pi}}\int_0^a e^{-\eta}\,d\eta. \tag{81} \]

At the center of the sphere, i.e. at the point \(R=0\),

\[ \frac{N(0;t)}{N_0} = 1+(k-1)\left[ \psi\left(\frac{R_0}{2\sqrt{Dt}}\right) -\frac{R_0}{\sqrt{\pi Dt}}e^{-\frac{R_0^2}{4Dt}} \right]. \tag{82} \]

To carry out numerical calculations by formula (82), it is necessary to know the value of the diffusion coefficient in the ionosphere, more precisely the value of the ambipolar diffusion coefficient, which depends on

mobility in it of electrons and ions. The question of diffusion in the ionosphere has at present still not been sufficiently investigated, so that exact values of \(D\) are unknown. Therefore the corresponding estimates are made by us for the smallest of the possible values of \(D_i\) for ions, which obviously gives somewhat overestimated values of the time of diffusion of the inhomogeneity. Using the gas-kinetic formula

\[ D_i \simeq \frac{1}{3}\,\bar v \Lambda \tag{83} \]

and substituting into it the value of the mean free path

\[ \Lambda = \frac{\bar v}{\nu_i} \tag{84} \]

and the value of \(\nu_i\)—the number of collisions of ions with neutral particles

\[ \nu_i^{(n)} \simeq \frac{4\pi\sqrt{2}}{3}\,a_0^2 n \bar v \tag{85} \]

and the mean velocity of the ions

\[ \bar v = \sqrt{\frac{8kT}{\pi m_i}} \simeq 6.2\cdot 10^5 \sqrt{T\,\frac{m}{m_i}}, \tag{86} \]

where the effective cross section of air is \(\pi a_0^2 \simeq 4.3\cdot 10^{-16}\ \text{cm}^2\), \(T\) is the temperature of the ionosphere, \(n\) is the density of neutral particles, and \(m\) and \(m_i\) are, respectively, the masses of the electron and the ion, we finally have that

\[ D_i \simeq \frac{3.8\cdot 10^{30}}{n}\sqrt{T\,\frac{m}{m_i}}\ \frac{\text{cm}^2}{\text{sec}}. \tag{87} \]

On substituting into (87) the values of \(T\), \(n\), and \(m_i\) recommended in the literature, one obtains the values of \(D_i\) given in Table II.

Table II

Values of the diffusion coefficient \(D_i\) for various layers of the ionosphere

Height \(Z\) in km	\(T\ ^\circ\mathrm{K}\)	\(n\ 1/\text{cm}^3\)	\(m/m_i\)	\(D_i,\ \text{cm}^2/\text{sec}\)
110. \((E)\)	300	\(8\cdot 10^{12}\)	\(1.7\cdot 10^{-5}\)	\(3.6\cdot 10^6\)
250 \((F)\)	1000	\(5\cdot 10^{10}\)	\(3.4\cdot 10^{-5}\)	\(7.8\cdot 10^8\)

From these values of \(D_i\), calculated by formula (82), tables of the function

\[ \frac{N(0,t)}{N_0}, \]

showing how the degree of ionization at the center of the sphere changes with time, have been calculated. The corresponding results

relating to the layers \(E\) and \(F\), are given in Figs. 11 and 12. In the figures, the ordinate gives the coefficient indicating the magnitude reached by the excess of the ionization of the sphere over the ionization \(N_0\)

Fig. 11. Results of the calculation for the \(E\) layer of the dependence of the value of the degree of ionization at the center of the sphere on time. Along the ordinate is plotted the value \(\gamma\), corresponding to the fact that the initial value of the excess ionization of the sphere over the surrounding medium \((k-1)N_0\) has, at time \(t\), become equal to \(\gamma(k-1)N_0\). The time \(t\) is given for different values of the radius of the sphere \(R_0\).

of the surrounding medium (equal initially to \((k-1)N_0\)), after a time \(t\), plotted along the abscissa in different scales corresponding to different values of \(R_0\) in meters.

From the figures it is seen that even at the center of the sphere the degree of ionization decreases in the layer \(E\) by a factor of 3–4 in \((40\div 50)\) sec, if \(R_0 \sim 200\) m, and in \((4\div 5)\) min, if \(R_0 \sim 500\) m. In the layer \(F\), however,

Fig. 12. Results of the calculation for layer \(F\) of the dependence of the value of the degree of ionization at the center of the sphere on time (see the explanations in the caption to Fig. 11).

the corresponding times are \(\sim 0.2\) sec and \((1\div 1.5)\) sec. Naturally, the peripheral part of the sphere will dissipate even faster*). Thus, during an experiment lasting 10–20 minutes,

*) We recall here that these estimates have been made for ions, so that the electron concentration will decrease still faster.

new scattering centers must arise in the reflecting region—some must be replaced by others. This process must be especially rapid in the layer \(F\). As for inhomogeneities of radius \(R_0 \sim 5000\) m, corresponding to the dimensions of the entire reflecting region as a whole, the degree of its ionization may decrease by a factor of \(3\)–\(4\) in the layer \(E\) in \(40\)–\(50\) minutes, and in the layer \(F\) in \(2\)–\(3\) minutes; thus, if an ionospheric layer has inhomogeneities of such a size, then the variability of the field at the earth’s surface can, with a certain approximation, be regarded in some cases as the result of the drift of a rough reflecting screen that changes little with time.

d) The case of absolute roughness of the reflecting region

Let us also consider the case when the reflecting region of the ionosphere is absolutely rough, i.e., consists of inhomogeneities whose linear dimensions are smaller than the wavelength. Such conditions may apparently occur at long wavelengths. In addition, rare cases are known when, even at short wavelengths, phenomena are observed indicating that the sizes of the inhomogeneities are small in comparison with the wavelength.

An absolutely rough surface, as is known, scatters isotropically—according to Lambert’s law, i.e., the value of the energy \(W(s)\), received within a given aperture of the angle \(d\alpha\), does not depend on \(\alpha\), but is proportional only to the value of \(d\alpha\), so that

\[ W(\alpha)=\operatorname{const}. \tag{88} \]

Therefore in formula (41) the function \(W(s)\) may be taken outside the integral sign. Moreover, since \(|s|>1\) has no real meaning (mathematically this means that these values of \(s\) are imaginary and give damped waves), in (41) the limits of integration \(\pm\infty\) may be replaced by \(\pm 1\), and as a result we obtain that

\[ \rho_R(\xi)= \frac{ \left\{ W(s)\displaystyle\int_{-1}^{+1} e^{\frac{i2\pi s\xi}{\lambda}}\,ds \right\}^{2} }{ \left\{ W(s)\displaystyle\int_{-1}^{+1} ds \right\}^{2} } = \left( \frac{\sin \dfrac{2\pi \xi}{\lambda}} {\dfrac{2\pi \xi}{\lambda}} \right)^{2}. \tag{89} \]

It follows from (89) that the correlation coefficient \(\rho_R(\xi)\) first becomes zero already at the distance \(\xi=\dfrac{\lambda}{2}\); since the sizes of the inhomogeneities \(\xi_0 \ll \lambda\).

Formula (89) makes it possible to check, from the experimental data obtained, whether the reflecting region is absolutely rough, and to estimate the dimensions \(\xi_0\) of the scattering inhomogeneities.

If an absolutely rough region does not change its shape and moves in the horizontal direction with constant velocity \(v_d\), and if, at the same time, changes in the amplitude \(R(t)\) of a single reflected signal are measured, then it is possible, from the results of measurements of \(R(t)\) at one point, to determine the drift velocity \(v_d\) of the ionosphere.

Indeed, in this case we have that the Doppler shift of the component of the field obtained from some roughness radiating to the point of observation at an angle \(\alpha\) is equal to

\[ f-f_0=\frac{2v_d\sin\alpha}{\lambda}. \tag{90} \]

Therefore the energy density of the wave beam is equal to

\[ W(s)=W(f)=W\left\{\frac{\lambda(f-f_0)}{2v_d}\right\}. \tag{91} \]

Taking into account that \(W(f)=W(s)=\mathrm{const}\), we obtain from (35)

\[ \rho_R(\tau)= \left\{ \frac{ \sin \dfrac{2\pi v_d\tau}{\lambda} }{ \dfrac{2\pi v_d\tau}{\lambda} } \right\}^{3}, \tag{92} \]

i.e., a formula analogous to (89), in which \(\xi\) is replaced by \(v_d\tau\). Comparison of (92) with the values of \(\rho_R(\tau)\), computed also with the aid of (32) from the experimental data, makes it possible to find the value of the drift velocity \(v_d\). It should be noted that in this case one can also estimate the linear dimensions of the inhomogeneities. Determining the time interval \(\tau_0\) for which \(\rho_R(\tau_0)\) has small values (say, as before, equal to \(e^{-1}\)), we obtain

\[ \xi_0 \simeq v_d\tau_0. \tag{93} \]

d) The disturbed state of the reflecting region and some of its indications

Above we have already analyzed the concept of a quiet or undisturbed ionosphere. It was indicated that this corresponds to the case when single reflected signals of the type shown, for example, in Fig. 13 are observed, in which the extraordinary and ordinary signals each constitute a whole, unsplit signal. However, fairly often, instead of such a simple pair of signals, one or two groups of signals are received. One such case is shown in Fig. 14, in which the extraordinary signal,

as is seen from the figure, apparently consisted, at the moment of filming, of three components that had overlapped one another, while the ordinary one consisted of a large number of signals, the mutual distance between which reached more than one hundred kilometers.

Fig. 13. Photo-oscillograms of doublets of signals reflected from the ionosphere: (extraordinary and ordinary signals). The time line is stretched between marks by 100 km.

An even broader group of signals is seen in Fig. 15, in which the extraordinary and ordinary “signals” constituted a single group of signals that filled a height interval of more than 300 km. If a cine recording of such a group is made, it can be established that it does not remain stable, and that within the given group the distribution of the amplitudes of the signals changes continuously already within a fraction of a second. This is seen, for example, from Fig. 16, in which the dynamics of one group, which “filled” a height region of the order of 100 km, was recorded successively (after \(1/6\) sec.).

From these examples it is clear that in such cases it is no longer possible to speak of a single reflecting region of the ionosphere; rather, one must state that during these periods the structure of the ionosphere makes possible the arrival at the receiving point of a multitude of “reflected” individual signals formed by various, one might say, “reflecting” regions. In those cases when the group is broad, one may conclude that such separate reflecting regions are removed from one another by distances much greater than the dimensions of a single reflecting region \((r_0 \sim 5\ \text{km})\). However, the overlapping of different signals in such a group, which is always observed, testifies to the fact that the reflecting regions forming them are located at almost identical distances from the observation point. This indicates,

Fig. 14. Photo-oscillogram of two groups of signals reflected from the ionosphere.

Fig. 15. Photo-oscillogram of a group of overlapping ordinary and extraordinary signals.

Fig. 16. Motion-picture recording of a group of signals that occupied a width on the oscillogram of the order of 100 km. The figure shows photographic prints of 27 consecutively taken frames, at a speed of 6 frames per second.

thus, on the fact that in these cases the layer consists of inhomogeneities of large size (at least having dimensions \(r_0\sim(5\div10)\) km, necessary for the formation of a single signal); moreover these inhomogeneities, apparently, change rather rapidly both their orientation and their position relative to the observation point, and at the same time adjoin one another. It may therefore be said that in such cases the reflecting part of the layer is, as it were, a strongly undulating surface, and the length of the longitudinal waves running along it (meaning the periodicity of the change in the electron concentration of the layer horizontally) is commensurate with \(r_0\sim(5\div10)\) km, while their depth (meaning the periodicity of the change in the electron concentration of the layer with height) also has dimensions of the same order. Such a process, varying in time, leads to the result that at the receiving point some “reflected” signals are rapidly replaced by others, formed by different regions moving over the observation point in different directions.

The examples given above may be regarded as signs of the disturbed state of the reflecting region of the ionosphere, and they are still far from exhausting all possible cases observed in a number of experiments. However, the literature still does not contain a sufficient number of results of reliable experimental investigations that would make it possible to systematize these data, to obtain a more or less detailed idea of the character of ionospheric disturbance, or to make any well-founded assumptions about the causes producing these phenomena. The question of the disturbed state of the ionosphere at the present time still cannot be set forth with the necessary completeness and in general lies beyond the scope of the present article. The data cited in this paragraph are purely illustrative in character. They make it possible to compare cases of a quiet and a disturbed ionosphere and to see how they manifest themselves in ionospheric investigations, the results of which are considered in this article.

§ 5. DRIFT OF THE REFLECTING REGION

a) Drift velocity of a reflecting region of unchanged “form”

In a number of experiments it was established, even before the data considered in §§ 2–4 were obtained, that horizontal motions—drifts—are observed in the ionosphere. The most commonly used method for determining the velocity vector \(\mathbf{v}_{\mathrm{d}}\) of these motions is the measurement of the variability of the amplitude \(R\) of a reflected unit signal at several (not fewer than three) separated points[^5].

If the distance \(\xi\) between them and the component of the drift velocity \(v_{\xi}\) are such that during the time \(\tau\sim\dfrac{\xi}{v_{\xi}}\) it does not have time to change appreciably,

the shape of at least part of the reflecting region of the ionosphere changes, then at different points curves \(R(t)\) of the same type must be recorded over certain time intervals, shifted, however, by a time

\[ \tau_{\xi}=\frac{\xi}{2v_{\xi}}, \tag{94} \]

since the velocity of displacement of the diffraction pattern at the earth’s surface is twice the drift velocity of the reflecting region. The latter is connected with the fact that the wave radiated upward undergoes a frequency shift twice, and with it a change of field at the receiving point: when it is incident on the moving roughness and when it is reflected from it. More rigorously, relation (94) follows from the formula determining the drift velocity \(v_n\), obtained from the correlation coefficient.

If we assume that the reflecting region of unchanged shape moves with velocity \(v_n\), then each component of the field arriving at the observation point at an angle \(\theta\) and scattered by some roughness has frequency

\[ f=f_0\pm \frac{2v_{\xi}}{\lambda}\sin\theta \simeq f_0\pm \frac{2v_{\xi}}{\lambda}\theta, \tag{95} \]

if \(\sin\theta\sim\theta\). Therefore the energy spectrum of this beam of waves is equal, for a chaotic distribution of angles (see (45)), to

\[ W(f)=W(\theta)= \frac{W_0}{\dfrac{2v_{\xi}\theta_0}{\lambda}\cdot \sqrt{2\pi}}\, e^{-\frac{\lambda^2(f-f_0)^2}{8v_{\xi}^{2}\theta_0^{2}}}, \tag{96} \]

where

\[ \theta_0^2=\frac{\overline{(f-f_0)^2}}{4v_{\xi}^{2}}\,\lambda^2. \tag{97} \]

Substituting (96) into (35), we obtain

\[ \rho_R(\tau)=e^{-\frac{16\pi^2 v_{\xi}^{2}\theta_0^{2}}{\lambda^2}}, \tag{98} \]

whence, for \(\rho_R\sim 1\), using (32), there follows the formula determining the velocity \(v_{\xi}\), namely

\[ v_{\xi}= \frac{\lambda\cdot \sqrt{\overline{|\Delta R|}_{\tau}}} {8\sqrt{\,4-\pi\,}\cdot R\theta_0}. \tag{99} \]

To compute \(v_{\xi}\) from formula (99), as we see, is practically impossible, since the value of \(\theta_0\) is unknown. However, if simultaneously

if observations are carried out at two separated points, then one can compute \(\theta_0\), equal to the value (see (48))

\[ \theta_0 = \frac{\lambda \overline{|\Delta R|_{\xi}}}{4\sqrt{4-\pi \xi R}} \tag{100} \]

for \(\alpha_0 \cong \frac{\pi}{2}\). It is obvious that the values of \(\overline{R}\) in formulas (99) and (100) are the same. Moreover, since \(\rho_R(\xi)\sim 1\), we also have \(\rho_R(\tau_\xi)\sim 1\). Consequently, substituting \(\tau=\tau_\xi\) in (99) and taking into account that \(\overline{|\Delta R|_\xi}=\overline{|\Delta R|_{\tau_\xi}}\), we obtain

\[ v_\xi=\frac{\xi}{2\tau_\xi}. \tag{101} \]

Observations at several points situated in different directions thus make it possible to determine the value of the vector velocity \(\mathbf{v}_{\text{d}}\) of the horizontal drift of the ionosphere. Of course, a more accurate and complete analysis of the results of such measurements requires the development of a method for processing experimental data which would make it possible to determine from the measurement results both the value of \(\mathbf{v}_{\text{d}}\) and the root-mean-square velocity of chaotic motions \(v_0\), as well as the ratio of the energy of the “specularly” reflected wave to the energy of the scattered signals—\(\beta^{3}\) (see § 2, a)—and the ratio of the energy of the “specularly” reflected wave to the energy scattered by the reflecting region as a whole owing to its drift, \(\beta_{\text{d}}^{2}\). Since the drift of the reflecting region gives yet another component of the field, and in expression (1), describing the field at the observation point, there appears the term

\[ a_{\text{d}}\cos(\omega_{\text{d}}t-\psi_{\text{d}}), \tag{102} \]

it is necessary to compute the corresponding formulas for the case when (102) is added to the right-hand side of equation (1). However, as has already been indicated, such a rigorous general method for processing measurement results has not yet been developed; therefore one has to confine oneself only to the results obtained in calculating \(v_{\text{d}}\) from measurements of the time shifts \(\tau_\xi\).*)

b) Results of measurements of drift velocity in the ionosphere

Within the range of the questions considered in this article, data on the velocity of horizontal motions are of interest, since the causes producing the statistical character of the ionospheric structure—the chaotic mobility and continuous change of

*) It should be noted that the method proposed in work 18 for determining \(v_{\text{d}}\) and \(v_0\) is qualitative and does not solve the problem indicated above.

Table III

Results of measurements of the velocity \(v_d\) of horizontal drift in the ionosphere

Layer	Limits of variation of \(v_d\), m/sec	Most frequently occurring or mean values of \(v_d\), m/sec	Prevailing direction of motion	Frequency in MHz	Period and place of observations
\(E_{\mathrm{spor}}\)	—	40 (in summer) 130 (in winter)	—	50–54	U.S.A. and Canada
\(F\)	80 ÷ 134	—	Northeast	5.8	Near Sydney, from XI.1947 to V.1948
	—	120	—	—	Europe, 1942 ÷ 1943
\(E\)	—	110	—	—	Canada
\(F\)	—	330	—	—	Canada
\(F\)	—	265	—	—	India
\(E_{\mathrm{spor}}\)	20 ÷ 110	50	Northwest	3–4	England
\(F\)	20 ÷ 110	50	Northwest	3–4	England
\(E_{\mathrm{spor}}\)	35 ÷ 58	—	—	50	U.S.A. (15 ÷ 16) V.1949
\(E\)	20 ÷ 200	90	Southwest Northeast	—	Germany (VIII ÷ X) 1942
\(F\)	20 ÷ 200	90	Southwest Northeast	—	Germany (VIII ÷ X) 1942
\(E\) and below	—	35	North-northeast	—	California, summer 1949 (from meteor trails)
\(F\)	83 ÷ 166	—	Northeast Southeast	—	In winter In summer \(\}\) India (IV.1948) ÷ (IV.1949)
\(E\)	20 ÷ 240	70	East (in summer) West (in winter)	2.4	England, (I.1949) ÷ (VI.1951)
\(E, F\)	20 ÷ 200	70	East (by day, in summer) West (at night, in summer) Southwest (in winter)	2.3	Washington, 1949 ÷ 1951

scattering centers. It may be assumed that these phenomena must in some way be governed by general motions—drifts of the ionosphere. Therefore, below are presented the results, published in the literature, of measurements of drift in different regions of the ionosphere.

The most complete measurements of \(v_{\mathrm{d}}\) have recently been carried out in works \(^{19}\) and \(^{20}\). In Fig. 17, for illustration, a curve is given for the distribution of the measured values of the velocities \(v_{\mathrm{d}}\) in the \(E\) layer at a frequency of \(2.4\ \text{MHz}\), obtained over a time interval of about two years. It is seen from the figure that the most frequently occurring value of the velocity was of the order of \(70\ \text{m/sec}\). The direction of the velocity vector, as

Fig. 17. Distribution curve of the values of the velocities \(v_{\mathrm{d}}\) of horizontal drifts observed in the \(E\) layer of the ionosphere at a frequency of \(2.4\ \text{MHz}\) in 1949–1951 in England.

shown by these experiments, varied during the day and from month to month. The corresponding polar diagrams given in this work have a very complicated character.

Figure 18 shows several daily monthly-mean polar diagrams of the variation in the magnitude and direction of the velocity vector. The numbers near the curves denote local time.

The results of the published data on measurements of \(v_{\mathrm{d}}\) are collected in Table III.

From Table III it is seen that the most frequently occurring velocities of horizontal drifts vary in different experiments within the limits

\[ v_{\mathrm{d}} \sim (35 \div 330)\ \text{m/sec} \tag{103} \]

and greatly exceed the value \(v_{0}\) of the chaotic motions in the reflecting region. However, the values of the angular spread \(\theta_{0}\) of the scattering centers of the reflecting region varied in different experiments within the limits \((0.5 \div 5)^\circ\), so that the component of the velocity \(v_{\mathrm{d}}\) in the direction toward the observer is equal to

\[ v_{\mathrm{d}}\sin\theta_{0}\sim v_{\mathrm{d}}\theta_{0}\sim (0.3 \div 30)\ \text{m/sec}, \tag{104} \]

i.e., it has values which, in order of magnitude, are comparable with the values of \(v_{0}\) (see § 2, b)).

Fig. 18. Polar diagrams of the velocity \(v_{\mathrm{d}}\) of horizontal drifts in the \(F\) layer of the ionosphere, obtained at a frequency of 2.4 MHz in 1950 in England.

§ 6. Brief Conclusion

The results considered in this article from a number of recent experimental studies of the ionosphere and their theoretical treatment show that the different layers of the ionosphere, in their normal, calm, undisturbed state, constitute something like a turbid medium. The character of its behavior is such that the electromagnetic field of radio waves reflected from it, emitted from the earth’s surface, as well as of radio radiation from cosmic sources passing through it[^21], is nonconstant and variable in time. The curves describing the behavior of the amplitude and phase of the field in time have a complex character, varying from point to point along the earth’s surface.

A successive analysis of the experimental results leads to the conclusion that within one zone of a layer—particularly the first Fresnel zone, which forms the reflected signal—there are many scattering centers in chaotic motion. If it is assumed that these scattering centers are something like isolated clouds, or some other regions of condensation of electron density, then it follows that their linear dimensions are of the order of \((400 \div 800)\) m, while the linear dimensions of the whole zone reflecting the radio signal are of the order of \((5 \div 10)\) km. Owing to rapid diffusion, such clouds must dissipate in several seconds in the \(F\) layer and in one or two minutes in the \(E\) layer. Therefore there must be a continuous replacement of such clouds, or, in other words, new regions of condensation and rarefaction of electron density must arise all the time. It can be shown that there is a chaotic distribution of the velocities of motion of these clouds, and it follows that the root-mean-square velocity is, in general, of the order of several meters per second. If, however, one starts from the assumption that the fine structure of the ionosphere is the result of the “running through” of condensations and rarefactions of ionization density, then these velocity values correspond to the velocity of propagation of longitudinal waves, and the cloud dimensions given above may be regarded as the wavelength of these waves. Under real conditions the picture is further complicated by the fact that the ionized regions drift with velocities sometimes reaching 300–400 m/sec and more.

Thus, the structure of the ionosphere has a statistical character; moreover, this “statisticalness” is one of its basic natural features, just as, for example, the thermal chaotic motion of molecules of a gas or liquid is a characteristic physical feature of these media. Naturally, it is important to study and determine what the basic mechanisms are that contribute to such behavior and structure of the ionosphere. However, up to the present there are still no definite data on this question. In the literature, certain phenomena have been considered which may contribute to the formation of inhomogeneities in the ionosphere and to the appearance in it

motions, in particular of a turbulent type. Since these phenomena have as yet been little studied from the theoretical side, and it has by no means been proved quantitatively what role they play in the ionosphere, it is not timely to analyze them in detail here and to give numerical estimates, which would show only that under certain conditions (as yet unknown) these phenomena, apparently, may influence the general state of the ionosphere. For the reader’s orientation we shall indicate only, in conclusion, some mechanisms of this type that have recently been considered.

It has been pointed out that plasma oscillations arising in it may be the cause of variability of the state of the ionosphere^14. Since the frequency of plasma oscillations which may arise near the region of the ionosphere forming the reflected wave is close to the frequency of the wave incident on it, the action of the longitudinal waves produced by them may be especially sensitive.

Next, the question of the influence of so-called magnetohydrodynamic waves in the ionosphere was theoretically considered^22. In analyzing the motions and dynamics of the ionosphere, low-frequency waves of this type may play the principal role; however, the conditions of their occurrence and their influence on the reflection of radio waves from the ionosphere are still unclear.

Recently an estimate has been made of the influence of acoustic energy generated by ocean waves on the ionosphere^23. It has been shown that this energy is sufficient for heating the ionosphere in the region, apparently, of the \(E\) layer and may lead to an appreciable change in the density at these altitudes. Other mechanisms have also been indicated, such as, for example, the influence of meteors, the influence of interstellar matter attracted by the gravitational field of the Sun^24, ^25, and other phenomena. However, all these considerations at present have only a speculative character. It is possible that the totality of such mechanisms determines the state of the ionosphere; therefore careful experiments will be required to demonstrate the influence of each of them.

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