THE INFLUENCE OF THE TROPOSPHERE ON THE STABILITY OF ULTRASHORT RADIO WAVE RECEPTION

B. A. Vvedenskii and A. G. Arenberg, Moscow

INTRODUCTION

1. In the present state of the question, the most probable picture of the propagation of ultrashort radio waves (u.s.w.) appears to be the following. In the very lowest layers of the troposphere there propagates a “terrestrial” component of the field, whose magnitude is determined mainly by the reflection of u.s.w. from the earth, by their diffraction, and by average refraction.

Within the horizon this component of the field undergoes a comparatively small effect from nonsteady refractive moments and is therefore relatively stable. The influence of refraction begins to manifest itself with certainty only in the immediate vicinity of the horizon, somewhat lengthening it and changing the phase relations between the direct and reflected (from the earth) rays. These questions were discussed in our preceding article1.

In the diffraction zone, and especially beyond the horizon, the influence of refraction is considerably stronger, the consequence of which is a relatively great instability of the field (see below). Diffraction theory, taking refractive phenomena into account, shows that in the diffraction zone average refraction decreases the field strength and does not increase it, as occurs in the zone close to the transmitter (the zone of geometrical optics)[^1a].

If the influence of the troposphere on the propagation of u.s.w. were limited to a comparatively insignificant monotonic decrease of the dielectric coefficient of air $\varepsilon$ with height (an “undisturbed” troposphere), then u.s.w. fields within the horizon (and, to a certain extent, even beyond the horizon) would differ comparatively little from those calculated by the usual nonrefractive formulas. Such fields would experience only slow and shallow variations, caused by comparatively small and almost regular changes in the vertical gradient (diurnal, seasonal, etc.).

However, in actual transmissions over such distances, u.s.w. fields experience considerably more rapid and deep changes than under the indicated idealized conditions. The totality of accumulated experi—

experimental data on the stability (or, more precisely, variability) of the u. s. w. field shows that the whole variegated picture of variations of this field (in what follows we shall call them “fadings,” which somewhat broadens the generally accepted interpretation of this term) can be subjected to numerical treatment only with very great difficulty.

1. Of the numerous attempts to classify fadings, the most complete should be considered the classification of Wainwright2. In receiving u. s. w. he distinguishes four main cases, shown in Fig. 1: a—

Fig. 1. Typical forms of fading. Measurements were made in England in the summer of 1939 (Wainwright).

absence of fading, which occurs very rarely; b—very rapid, normally shallow fading (with periods of about 30 sec.); c—slow fading (with a period of about 5 min.), accompanied by deep changes of amplitude, and, finally, d—a rather frequently occurring case of the simultaneous existence of slow and rapid fading.

In addition, as a number of authors indicate, in receiving u. s. w. over relatively large distances there sometimes occur such strong attenuations of the field that reception is temporarily interrupted completely.

In Fig. 2 are given the results of processing the experiments of England, Crawford, and Mumford3, who experimented (within the horizon) on waves of 4.7, 4, and 2 m. From these curves (with a certain allowance, their character may also be extended to some distance beyond the horizon, i.e., to part of the diffraction zone) it is seen that, independently of wavelength, the values of the mean field (heavy lines) lie near zero db, which, owing to the chosen scale, corresponds to the field of “free space” (isolated dipole). The vertical strokes correspond to the greatest and smallest values of the field for each day.

For practice, the so-called “stability characteristic of communication” is important, i.e., a curve showing what percentage of the total time the field was equal to, or less than, the ordinate indicated. Such a characteristic for the experiments indicated is given in Fig. 3. It is seen from it that the average (in time) field only slightly exceeds zero db, and this value differs very little from that obtained (by calculation) from the conditions of the locality and the arrangement of the transmitting and receiving antennas.

Fig. 2. Averaged curves of the UHF field. Horizontal polarization (England, Crawford, and Mumford)

Other experiments and calculations show that the introduction of the equivalent radius \(a_e\) gives a value closer to the average field. Thus, within the horizon the mean refraction changes the field strength only slightly and, moreover, the time-average value of the field agrees quite well with the calculated one. But alongside this it becomes clear that it is necessary to introduce some new factor which would make it possible, with sufficient freedom from artificial assumptions, to explain the observed “swings” of the field about its calculated (which are also its mean) values; in other words, to explain the presence of fadings.

Fig. 3. Stability characteristic of reception, showing what percentage of time the field was equal to or less than the indicated ordinate (England et al.)

1. The cause of fadings at the present time most authors are inclined to regard as the presence of inhomogeneities of the troposphere sufficient for reflection of UHF. These inhomogeneities may have the character of rather sharply bounded layers or of some other formations (for example, “globules”), reflection from which may be either diffuse or specular in character.

In recent years many papers have been devoted to the study of these inhomogeneities. The earliest authors, in particular Watson Watt and his collaborators \(^{4}\), tried in every way to prove the ionic nature of these

inhomogeneities, but from this point of view they should be regarded as manifestly insufficient. As we shall see below, in the troposphere there exist inhomogeneities of density, temperature, and aggregate state of the water contained in the air. These inhomogeneities prove sufficient for jumps in the dielectric coefficient of order \(10^{-4}—10^{-6}\) to appear in the troposphere, as a result of which fairly intense reflection of ultrashort waves may occur.

The existence of such layers is proved both aerologically and by a large number of special experiments, which confirm oscillographically the reflection of radio waves, including ultrashort waves, from the troposphere at heights of the order of several hundred meters and higher. It is obvious that such inhomogeneities must possess the same degree of instability that is inherent in the outlines of the lower edges of clouds. As a result, reflection from such inhomogeneities must also be unstable.

In the presence of such reflection, at the receiving point there occurs interference of the “ground” component of the field mentioned above with its “tropospheric” component. This interference is what gives rise to the observed sharp deviations of the mean field values from the values determined by taking refraction into account in an “undisturbed” troposphere, i.e., fadings. This also, apparently, explains the rare cases of propagation of ultrashort waves over distances exceeding the distance to the horizon by several times.

Such a conception is confirmed by experiments with the propagation of ultrashort waves over various distances. At relatively small distances, when the angles of incidence of the ultrashort waves on the reflecting inhomogeneities of the troposphere are sufficiently large, the reflection coefficients are small. In this case, at the receiving point the ground component of the field predominates and the reception is relatively stable. But as the distance increases, the angles of incidence decrease and the influence of the “roughnesses” of the individual inhomogeneities must become appreciable. From the standpoint of the influence of roughnesses, the situation here is essentially the same as in grazing reflection of ultrashort waves from the ground, or in the reflection of light from a matte plate when incident at angles close to \(90^\circ\).

In this case the field of the tropospheric component increases, and since, along with this, the field of the ground component decreases (because of the increase in distance), the resultant field naturally becomes less stable. In this situation the reception may be sufficiently intense. Finally, at still greater distances the field of the ground component becomes quite insignificant, and practically only one tropospheric component remains at the receiving point, which by its nature cannot be stable and in general does not always exist.

Developing this conception, it is easy to arrive at conclusions about the dependence of fadings on distance, wavelength, time of day, season, etc.; some quantitative conclusions are also possible. However, there is no doubt that the experimental data relating to this are still too few, and that in further work their number must be supplemented in every possible way.

4. The picture presented of the structure of the troposphere, in which the monotonic decrease of the dielectric coefficient of air is cut through by layers

of sharp inhomogeneities is probably only a crude model of reality. One must suppose that, in the normal state of the troposphere, the change of the dielectric coefficient with height is represented by some more or less winding curve; in the particular case when individual bends of this curve become very abrupt, a picture of “layers” appears. Such an idealization is quite acceptable. The qualitative mechanism of the variability of the field in time may be imagined as the result of a change, in time and space, of the height curve (it would be better to say—the surface) of the dielectric coefficient.

The present article is an attempt to analyze and critically compare works concerning the propagation of ultrashort waves in a “disturbed” troposphere, when sharp inhomogeneities unstable in time are present.

Questions connected with the role of the ionosphere in the propagation of the long-wave end of the ultrashort-wave range stand apart. These questions could constitute the content of a separate article.

§ 1. HYPOTHESES ON THE EXISTENCE OF IONIC LAYERS IN THE TROPOSPHERE AND THEIR UNTENABILITY

1. Beginning approximately in 1935, numerous articles began to appear in the literature devoted to the study of the reflection of radio waves from low layers of the atmosphere. In these articles, the results of investigations by means of sounding the atmosphere with short-duration radio pulses were presented and discussed1.

Thus, Mitra and Syam[^6], who studied the reflection of frequencies from 1 to 6 MHz from an ionized layer situated at heights of the order of 90—110 km (the \(E\) layer), noted diffuse reflection from a height of the order of 55 km (the \(D\) layer). Pointing to the absorbing action of this layer, they recalled the existence in the atmosphere of an ozone layer having maximum density at heights of the order of 30—50 km, considered by Chapman[^7].

Colwell and Friend[^8], working at frequencies of 1.6 and 3.49 MHz with pulses of 10 μsec, obtained reflection from effective heights of 5—30 km and 40—55 km. At the same time they noted the instability of such reflections, manifested especially sharply at sunrise and sunset.

Watson Watt, Bainbridge-Bell, Wilkins, and Bowen[^9], who observed reflection from heights of 6—60 km at frequencies of 6—12 MHz, came to the conclusion that there existed three reflecting layers situated at heights of the order of 6—14 km, 15—50 km, and 60 km. Simultaneously with these authors, Mitra[^10] published a note in which he pointed out that, in explaining the reflection of radio waves from heights less than 50 km, one can hardly proceed from the usual conceptions of ionized

layers. The layers corresponding to these heights he proposed to denote by the letter \(C\).

In reply to this note, Colwell, Friend, Holme, and Hill\({}^{11}\) indicated that they saw their merit chiefly in extending the existing notions about \(C\) layers, which can sometimes descend to heights of the order of \(1—5\) km. The method of these observations, carried out at frequencies of 1.61, 2.39, and 3.49 MHz, was described in an article by Colwell and Friend\({}^{12}\). Reception was carried out on a loop rotating in the vertical and horizontal planes. This loop was set up in such a way that the direct signal, received directly from the transmitter, was strongly weakened.

Table 1 gives, in part, the results of measurements of the heights of the lower layers obtained by these authors.

Table 1

Date	Time	Height \(C\), km	Height \(C_1\), km	Date	Time	Height \(C\), km	Height \(C_1\), km
5/III 1936	17,00	5	17	30/V 1936	11,40	3,6	10
9/III 1936	18,30	3—12	35	6/VI 1936	11,10	3,7—4,6	18
11/III 1936	20,30	11	35	6/VI 1936	23,29	7,5	20
12/III 1936	14,56	8—12	—	7/VI 1936	2,55	8,6	22
31/III 1936	14,57	7,5	18	7/VI 1936	00,35	4,3	16
6/IV 1936	21,20	10,5	18	7/VI 1936	23,03	8	18
28/IV 1936	11,15	2—4	12,14	8/VI 1936	17,30	4,3	14,5
10/V 1936	00,03	4,3	12	11/VI 1936	17,15	0—2	16
24/V 1936	13,05	5,1	12	29/VI 1936	00,30	3,5	21

From this table it is evident that the heights of the \(C\) and \(C_1\) layers vary within rather wide limits. The authors mentioned also noted large changes in the intensity of the signals reflected from these layers. Definite sharp reflections from the troposphere were also observed by Bakshit and Bhargava\({}^{13}\), who worked in Calcutta at frequencies from 1 to 15 MHz.

1. In the middle of 1937, Watson Watt, Wilkins, and Bowen\({}^{4}\) published the results of their investigations, carried out at frequencies of 6—12 MHz. These authors believed that the sounding pulses undergo (in vertical radiation) several reflections from four ionized layers \(ABCD\) (see the corresponding lines in Fig. 4). The lines denoted by these letters with subscripts 2, 3, 4, etc., correspond to fictitious equivalent heights obtained in repeated reflections from the layers \(ABCD\). Points situated on one and the same horizontal line belong to one definite oscillogram. In processing these results, Watson Watt et al. came to the conclusion that the individual reflections are grouped around equivalent heights that are approximately multiples of 8.39, 9.33, 10.26, and 10.76 km.

However, a sufficiently cursory glance at the set of points in Fig. 4 is enough to become convinced of the obvious tendentiousness of this interpretation. In fact—

tively, the heights, i.e., in fact the delay times of the individual reflections, are distributed almost uniformly over the entire scale of equivalent heights. In presenting the results of the processing of other experiments, the authors mentioned

Fig. 4. Equivalent heights of the reflecting layers (Watson Watt et al.)

give the values of the equivalent heights of the reflecting layers as about 8.5; 9.3; 10.3; 10.75 and 13.5 km. Reflection from lower heights was not observed by them, since this was hindered by the relatively great duration of the pulses (20 μsec.).

Analyzing the oscillograms of individual soundings, Watson Watt et al. found that the ratio of the “amplitude” (i.e., the deflection on the oscillogram) of a pulse that had undergone five reflections from a given layer to the intensity of a pulse that had undergone one reflection from it is of the order of 0.2. On this basis, and assuming that each subsequent reflection reduces the intensity of the pulse in accordance with the reflection coefficient \(\Phi\), they determined this reflection coefficient (for each of the layers) as \((0.2)^{1/4}=0.67\). In the concluding part of their paper Watson Watt et al. give the value \(\Phi=0.7\).

It is obvious that one could agree with such a method of determining the reflection coefficient only in the hypothetical case of sounding the atmosphere not by a spherical wave (whose energy decreases inversely proportional to the square of the distance), but by a plane wave, with a completely reflecting (ideally conducting) earth.

1. The fallaciousness of all these conclusions becomes especially obvious if one determines what electron (or ion) concentration would have to exist in the lower layers in order that frequencies of the order of 10 MHz should be reflected at least somewhat regularly from these layers with a reflection coefficient equal to 0.7.

In fact, in the case of normal incidence of a wave on an ionized layer (which we shall regard as sharply bounded; for a non-sharp layer the conditions would be still more stringent) the modulus of the reflection coefficient, as is known, is determined by the expression

\[ \Phi^2=\frac{(n-1)^2+k^2}{(n+1)^2+k^2}, \tag{1} \]

where

\[ 2n^2=\sqrt{\varepsilon^2+2\sigma^2T^2}+\varepsilon \quad\text{and}\quad 2k^2=\sqrt{\varepsilon^2+2\sigma^2T^2}-\varepsilon, \]

where \(T\) is the period of oscillation.

For the dielectric constant of the ionized gas and its electrical conductivity \(\sigma\) in the case of fields varying according to harmo-

nical law, as is known, we have:

\[ \varepsilon = 1-\frac{4\pi Ne^2}{m(\omega^2+\nu^2)} \quad \text{and} \quad \sigma=\frac{Ne^2\nu}{m(\omega^2+\nu^2)}; \tag{2} \]

here \(e\) and \(m\) are the charge and mass of the electron (or ion); \(N\) is the number of “charges” (electrons or ions) per cubic centimeter; \(\nu\) is the mean number of collisions experienced by a “charge” in 1 sec.

Combining these expressions, we obtain:

\[ 1-\varepsilon=\frac{4\pi\sigma}{\nu}=2\sigma T\frac{\omega}{\nu}. \tag{3} \]

For the heights of interest to us, of the order of \(10\) km, the air pressure is \(\sim 160\) mm and its temperature is \(\sim 220^\circ\) K. Under these conditions \(\nu\) may be taken as \(\sim 1.3\cdot 10^{11}\ \text{sec}^{-1}\) for electrons and \(\sim 6\cdot 10^{9}\ \text{sec}^{-1}\) for ions. Thus, for frequencies of the order of \(10\) MHz the relation \(\nu \gg \omega\) is valid; substituting this into expression (3), we find that \(1-\varepsilon \ll \sigma T\). Consequently, it may be assumed that

\[ 2n^2=\sqrt{1+4\sigma^2T^2}+1 \quad \text{and} \quad 2k^2=\sqrt{1+4\sigma^2T^2}-1. \]

But then

\[ n=\frac{1+\Phi}{1-\Phi}, \tag{4} \]

which, for \(T=10^{-7}\) sec. and \(\Phi=0.7\), gives the value of the conductivity \(\sigma=8\cdot 10^7\) CGSE. Introducing this value into formula (2) and assuming that the layer under consideration consists of electrons, we determine the electron concentration corresponding to the found value of \(\sigma\) as

\[ N=\frac{m\nu\sigma}{e^2}=4.11\cdot 10^{10}\ \text{electrons}/\text{cm}^3. \]

Of course, in the case of ions the values of \(N\) are still larger1. Watson Watt et al. found that \(N\) is of the order of \(5\cdot 10^{12}\ \text{ions}/\text{cm}^3\), and explained the possibility of the existence of so large an \(N\) by the action of thunderstorms. However, at least in the absence of thunderstorms, the greatest values of ion concentration at these heights, well studied in various flights, can hardly exceed several thousand ions/cm\(^3\) (see point 6 of the present paragraph). Therefore the attempt by Watson Watt et al. to explain the (stable!) reflections they observed by the presence of low ionized layers is completely unacceptable.

1. Soon after the paper by Watson Watt there appeared a paper by Friend and Colwell[^15], devoted to the investigation of reflecting regions in the troposphere

at frequencies 1.61; 2.39; 3.49 and 7.49 MHz. These authors carried out reception on a single-turn loop (see above), by rotating which it was possible to obtain equal amplitudes of the direct and reflected pulses; under these conditions the passage conditions of both pulses through the receiver become identical. In observations of the peaks of the pulses it was possible to read time intervals down to 1.5 μsec, which made it possible to study reflection from the very lowest layers.

Observations were carried out for a year and a half. It was noted that with an increase in atmospheric pressure the height of layer \(C\) decreased, and vice versa. Fig. 5 gives a sketch of an oscillogram obtained at night on 1.VIII.1937. Each square corresponds to a height of 1.515 km. The central pulse \(C_1\) corresponds to a reflection height of about 0.78 km, the pulse \(C_2\) to a height of about 2.56 km, and the pulse \(G\) to a reduced direct signal from the transmitter. Some broadening of the pulses may be interpreted as a consequence of the “diffuseness” of the lower boundary of the reflecting layers.

Fig. 5. Reflected pulse before an auroral flare (Freund and Colwell)

It should be noted that Colwell and Freund, in this paper as well, do not express definite judgments about the nature of the reflecting layers, limiting themselves only to a vague indication of the connection of the processes considered with solar activity and magnetic disturbances.

They expressed clearer judgments on this matter considerably later, in another paper of theirs\(^{16}\), which we cite in the following paragraph.

1. The work of Watson Watt et al.\(^{4}\) in 1937 was subjected to serious criticism by Appleton and Piddington\(^{17}\). From the observations of these authors, the large values of the reflection coefficient (0.7) obtained by Watson Watt et al. were not confirmed. Appleton and Piddington determined the reflection coefficient by comparing the magnitudes of the direct and reflected pulses (allowing for the attenuation of the latter over the path it had traversed). The values they obtained for the reflection coefficient for the layer at an altitude of 10 km were of the order of \(10^{-4}—10^{-5}\), i.e., many times smaller than those of Watson Watt.

Discussing these results, Appleton and Piddington put forward the supposition that the probing pulses may be reflected not from layers, but from certain chaotically arranged inhomogeneities, playing the role of scattering centers, separated from one another by distances \(R \gg \lambda\). The question of the nature of such inhomogeneities is scarcely touched upon in this paper, and the impression remains that the authors themselves do not clearly distinguish the nature of layers located at heights of the order of 5–15 km and 50–60 km.

Denial of the possibility of the existence of ionized layers at heights of the order of 10 km is also contained in Piddington’s paper\(^{18}\), which takes the value of the reflection coefficient at a frequency of 10 MHz to be \(2 \cdot 10^{-5}\).

Since in this case the value \(\sigma T\) cannot be very large, on the basis of formulas (1) and (3) he obtains approximately that

\[ n=1+\frac{\sigma^{2}T^{2}}{2} \quad\text{and}\quad k=\sigma T, \]

whence

\[ \Phi \simeq \frac{\sigma T}{2}. \tag{5} \]

For \(\Phi=2\cdot 10^{-5}\) and \(T=10^{-7}\) sec., this expression gives the value \(\sigma=400\) CGSE, corresponding to an ionic concentration \(N=6\cdot 10^{8}\) ions/cm\(^3\), which appears quite incredible (see below). Therefore Piddington denies the possibility of the existence of low ionized layers and regards the reflected pulse as the result of scattering of energy by individual ionic aggregations (globules).

Analyzing the possibility of such an explanation and assuming that the dimensions of such globules are small in comparison with the wavelength, he starts from the formula:

\[ \Phi=\frac{2Q}{r}\cdot\frac{e^{2}}{mc^{2}}\cdot\frac{\omega}{\nu}, \tag{6} \]

where \(\omega \ll \nu\) (here \(Q\) is the total number of charges; \(r\) is the distance from the globule to the observer).

Substituting into this formula the numerical values of all the quantities, for a globule of diameter \(10\) m he obtains \(Q=2\cdot 10^{2}\) ions, which corresponds to an ionic concentration \(4\cdot 10^{11}\) ions/cm\(^3\), i.e. considerably greater than was obtained in calculating reflection from a plane ionic layer. On this basis, Piddington also rejects this hypothesis, and with it the entire “ionic conception” of the low layers. He gives arguments that make it possible to interpret the reflection of sounding pulses from low altitudes by the presence of layers whose dielectric coefficient changes sharply with height but is not of ionic nature.

1. Especially vigorous opponents of the ionic explanation of reflection from the troposphere are Gish and Booker\(^ {19}\), who summed up all the previously adduced objections and added a number of new ones. Pointing to the doubtful nature of obtaining a false echo because of defects in the apparatus, they nevertheless fully admit the possibility of obtaining an echo as a result of reflection from some terrestrial objects and recommend that this be clarified.

As we shall see later, this question was to a certain extent investigated by Harang and Stoffregen\(^ {20}\) in Tromsø.

Gish and Booker devoted their entire study to proving that neither in the troposphere nor in the lower stratosphere can there exist any appreciably intense ionization. Referring to Mitra\(^ {10}\), Appleton and Piddington\(^ {17}\), and a number of other authors, they cite the results of direct observations in the mountains and on the stratosphere balloon “Explorer-II,” which rose to an altitude of \(22\) km (North Dakota, 11/II 1935). During this flight the greatest ionic concentration (at an altitude of \(14.8\) km) was only \(5\,300\) ions/cm\(^3\).

This value of \(N\) is approximately \(10^{7}\) times smaller than that obtained by Watson Watt et al.\(^4\). However, as an even weightier argu-

ment, Gish and Booker give a calculation of the energy necessary for the creation and maintenance of a low layer of ionization sufficient for the reflection of sounding pulses.

Proceeding from the value of the energy necessary to form an ion in air (about 35 eV), and from the recombination coefficient \(a_0=1.6\cdot10^{-6}\) (at \(760\ \mathrm{mm}\ \mathrm{Hg}\) and \(273^\circ\mathrm{K}\)), as well as from the mean temperature of the stratosphere, for the power necessary to maintain ionization at an altitude of \(11\ \mathrm{km}\), they obtained the value \(P=3.4\,N^2\cdot10^{-24}\ \mathrm{W}/\mathrm{cm}^3\). Thus, for the value \(N=5\cdot10^{12}\ \text{ions}/\mathrm{cm}^3\), indicated by Watson Watt and others, it turns out that \(P=84\ \mathrm{W}/\mathrm{cm}^3\). True, in this calculation the role of free electrons is not taken into account; taking them into account considerably lowers the value of \(P\); therefore the quoted value of \(P\) should be regarded as an upper limit.

As a lower limit of \(P\) one obtains a considerably smaller value, \(P=8.6\cdot10^{-7}\ \mathrm{W}/\mathrm{cm}^3\), but even this value is still astonishingly large in comparison with the power of solar radiation, which at the upper boundary of the earth’s atmosphere may be taken as equal to \(1\,340\ \mathrm{W}/\mathrm{m}^2\). Therefore Gish and Booker arrive at the decisive conclusion that other hypotheses must be invoked to explain the causes of any more or less regular reflection of radio waves from the troposphere.

However, accidental sporadic reflections of ultrashort waves due, apparently, to ionization are nevertheless possible. This was pointed out, for example, by N. D. Papaleksi and A. N. Kazantsev. The character of the reception of ultrashort waves during auroras, noted by Ferrell\(^{21}\), also apparently confirms the possibility of such reflection. It turns out that during auroras there appear fluctuations of the field even of nearby stations, whose fields usually do not fluctuate. In addition, a hoarseness of tone appears which is not observed in ordinary fadings. Ferrell is inclined to explain this circumstance by the formation of ionic accumulations of high concentration, whose rapid displacement causes a change in frequency as a result of the Doppler effect.

§ 2. CAUSES OF THE REFLECTION OF RADIO WAVES FROM THE TROPOSPHERE

1. Aerological investigations of the lower layers of the atmosphere show that the distribution of temperature and humidity with height may vary within rather wide limits\(^{1}\). At the same time, at altitudes from \(0.5\) to \(3\ \mathrm{km}\), inversions very often occur (by this term we understand not only an increase of temperature, but also any noticeable decrease in the rate of its fall with height).

On summer days, with intense heating of the earth by the sun, inversions are usually absent; in winter, however, they may persist rather stably. Such inversions may be accompanied by rather strong changes in humidity, whose influence on the magnitude of the dielectric coefficient of air was considered in our preceding article\(^{1}\).

Water contained in the atmosphere may also be found in other aggregate states. The transition of vapor into the liquid state (formation of clouds—

\(^{1}\) See, for example, in Humphreys\(^{22}\), Wegener\(^{23}\), Palmén\(^{24}\), and others.

must be accompanied by a considerable change in \(\varepsilon\), for for the liquid phase \(\varepsilon\) is of the order of 80. The further transition of water into the solid state, on the contrary, must lead to a decrease in \(\varepsilon\), since for ice \(\varepsilon \simeq 2.7\).

In those cases where changes in the humidity of the air, or in the aggregate state of the water present in it, occur in such a way that a sufficiently sharp (in comparison with \(\lambda\)) inhomogeneity is formed, the resulting jump in \(\varepsilon\) may prove sufficient for the reflection of radio waves. The layers thus produced may be either visible (clouds, fogs) or so thin as to remain invisible.

In discussing this question, Paddington\(^{18}\), on the basis of data from aerological investigations of the Kew Observatory\(^{25}\), pointed out that out of 47 aeroprobe flights, in 17 saturation was found (at least at one altitude); in the remaining 30 soundings the relative humidity at one or several altitudes was \(90\%\).

If it is assumed that in the troposphere there always exists some saturation layer in which water has fallen out in the form of droplets, and if it is considered that the additive law is applicable in determining \(E\) of an emulsion of water in air\(^{1}\), then this value of \(\varepsilon\) may be approximately determined from the formula

\[ (\varepsilon - 1) q \simeq 80 q_{\mathrm{H_2O}}, \]

where \(q\) is the weight of a unit volume of the emulsion, and \(q_{\mathrm{H_2O}}\) is the weight of the water contained in the same volume.

In the case of ice this formula becomes somewhat more complicated.

Having obtained the value of \(\varepsilon\), it is easy to find the coefficient of reflection of radio waves from an emulsion layer, whose surface is assumed to be sharply bounded. For normal incidence, for this coefficient of reflection we have approximately:

\[ \Phi = \frac{n - 1}{n + 1} = \frac{\varepsilon - 1}{4}. \tag{7} \]

According to Paddington’s calculations, carried out according to the indicated scheme, the values of \(\Phi\) for layers lying at heights from 2 to 10 km cannot exceed \(10^{-4}\). However, such an approach to the determination of \(\Phi\) is too primitive, since the reflection of radio waves from an emulsion layer must have a diffuse character (see below).

1. The first to draw attention to the influence of the stratified structure of the troposphere on the propagation of ultra-short waves was apparently R. Khell\(^{26}\), whose work has been repeatedly cited by many authors. Analyzing experimental data relating to the propagation of waves of length 5 m over a distance of 160 km (exceeding the distance to the horizon by approximately a factor of 5), he noted an increase in reception in the presence of temperature inversions at a height of about 1000 m, and a weakening of reception when masses of polar air intruded.

Some data illustrating the influence of inhomogeneities of the troposphere on the propagation of ultra-short waves are also contained in the article

\(^{1}\) More rigorous calculations, in view of the general hypothetical nature of the whole reasoning, would hardly be meaningful.

Scholte and Egersdörfer[^27]. Field records obtained by these authors in reception at distances of 70 and 225 km give a correlation between the presence of inversions and strong reception.

Without attaching special significance to these experiments, we nevertheless consider it necessary to emphasize their scale, characterized by simultaneous aerological observations along a very long path.

A major step forward was the work of England, Crawford, and Mumford[^28], who investigated the stability of reception on wavelengths from 1.6 to 4.8 m at a distance of 113 km (transmission over the sea).

Analyzing these observations, the authors came to the conclusion that the received field has an interference structure. To test their assumption they used a method of radio transmission with a varying frequency, analogous to the method of Appleton and Barnett[^29] for studying the ionosphere. To vary the frequency of the transmitter, operating on a wavelength of 4.54 m (66 MHz), a short-circuited turn rotating in the transmitter circuit was used. The resulting frequency variation reached 6.2 MHz, and the pass band of the receiver was about 3.2 MHz. The observations were recorded with the aid of a motion-picture camera and an oscilloscope, whose sweep frequency was synchronized with the speed of rotation of the turn in the transmitter.

Fig. 6. Interference structure of the field (England et al.)

Figure 6 gives five frames from a film, the filming of which lasted 95 sec. The ordinates of the curves are proportional to the square of the field intensity. The characteristic of the receiver is shown by a dotted line.1 The presence of several maxima and minima (chiefly the latter) on these curves gives grounds for supposing that the received field is the result of the interference of several components differing from one another in phase.

Fig. 7. Such an appearance would be taken by the curves of Fig. 6 if there were no cutting action of the frequency characteristic of the receiver

In radio transmissions over distances exceeding the distance to the horizon, one of these components should be understood as the field at the receiving point obtained as a result of the combined action of diffraction and refraction (the mean value). In the case of distances lying within the horizon, this component may, in simplified fashion, be regarded as the result of ordinary interference of the direct and ground-reflected rays. The remaining components should be understood, independently of

distances, fields obtained as a result of reflection of ultrashort waves from the indicated tropospheric layers.

The further development of these works was described by Englund, Crawford, and Mumford[^30]. In Fig. 8 three series of oscillograms are presented, obtained by them under different atmospheric conditions. The number of components corresponding to each of these series was determined by Englund et al. on the basis of the form of these records. Proceeding from the difference in frequencies corresponding to two neighboring minima \((\Delta f)\), they found that the difference in path between two components

\[ \left(\Delta r = \frac{c}{\Delta f}\right) \]

may lie within the limits from several meters to 550 m. In the case under consideration such a path difference corresponds to reflection from layers situated at heights up to 5 km1.

Fig. 8. Records of the field for different numbers of components. The time of each record is indicated on the curves (Englund et al.)

In the cited article are given the results of comparing the heights of reflecting layers, determined (for those cases when the field could be considered as the result of interference of two components) by means of the indicated processing of the oscillograms, with the results of simultaneous aerological observations2. The experimental points are grouped fairly well about those heights which are characterized by abrupt changes. The latter were calculated on the basis of aerological data on the temperature and humidity of the air.

As we shall see from what follows (see § 3), the values of \(\Delta \varepsilon\) calculated by the cited authors for different air masses differing in temperature and humidity3 are quite sufficient for the reflection of ultrashort waves from these layers.

1. The question of the height of reflecting tropospheric layers and their nature is considered in considerable detail in the work of Friend and Colwell[^16]. In this article they for the first time spoke against the hypothesis of reflection of sounding

pulses from low ionized layers and joined in explaining these reflections by inhomogeneities of the air of a meteorological character, which is connected with changes in the dielectric coefficient of air.

Having organized special flights to measure the temperature of the lower layers of the air, they simultaneously measured the height of the reflecting layers by the method of sounding pulses. The duration of these pulses was in the range from 4 to 10 μsec.; the wavelength was equal to 125.2 m.

In Figs. 9a and 9b are given the curves of the distribution of temperature with height, obtained during these flights. The horizontal arrows correspond to the heights of reflection.

Fig. 9a. Case of two reflecting layers. Temperature measured at 17 h. Height measured at 18 h (Friend and Colwell)

Fig. 9b. Case of a sharp change of temperature at a height of 1.55 km: \(c_1\)—very weak intermittent reflection, \(c_2\)—strong fluctuating reflection (Friend and Colwell).

From these graphs it follows that the presence of inversion layers was constantly accompanied by reflection from these layers. Of greatest interest is Fig. 9b; it corresponds to the case of a sharp jump in temperature, accompanied by the formation of clouds. As we see, the height of the reflecting layers in this case underwent rather noticeable oscillations, indicating the instability of the reflection and its diffuse character.

During a thunderstorm, according to Friend and Colwell, during 5–15 sec. after a lightning flash the height of the reflecting layers increased, and then slowly fell to its initial value. It may be supposed that this phenomenon is connected with thermal processes in the troposphere; however, this question is still unclear.

Colwell and Friend also published a note \(^{33}\), in which a comparison is given of the heights of reflecting layers determined both by radio sounding at frequencies from 1.614 to 17.31 MHz and by measurements on an airplane. The results of these observations (part of which correspond to the data of Figs. 9a and 9b) are given in Table 2. The height of the locality above sea level was 290 m.

The agreement of the heights of the reflecting layers, determined by these two completely independent methods, proved to be very good. Change—

their frequency within the limits indicated above did not give a noticeable difference in the height of the reflecting layers. This points to the possibility of the existence of rather sharp boundaries between individual layers of air. Such a result appears quite unexpected, for it would seem that for ultrashort waves the other case should be typical, when the boundaries of the layers are blurred and reflection from them has a diffuse character.

Table 2

Flight	Date	Height of the reflecting layer (above sea level) according to radio measurements, km	Height of temperature inversions (above sea level) according to flight data, km
1	22/XII 1938	1.4	1.2
2	28/XII 1938	1.09	1.04
3	2/I 1939	1.2 and 1.8	1.15 and 1.7
4	5/I 1939	from 1.52 to 1.9	1.5
5	6/I 1939	1.14; 1.29; 1.50; 1.63	1.16; 1.50
6	22/IV 1939	1.45; 1.75; 2.3; 2.6	from 1.4 to 1.65; 2.1

Very recently, Freind\(^{34}\) published the results of experiments carried out by Harvard University jointly with the Weather Bureau. The air temperature and its humidity were determined with the aid of radiosondes. On the basis of these data the values of \(\varepsilon - 1\) were calculated by the formula\(^{1}\)

\[ (\varepsilon - 1)10^{-6} = \frac{157.5}{T}\left(p_{\text{air}} + \frac{4800}{T}p_{\text{water vapor}}\right). \]

Here \(p_{\text{air}}\) is the air pressure and \(p_{\text{water vapor}}\) is the vapor pressure of water, given in millibars; \(T\) is the absolute temperature.

Since the radiosonde readings were insufficient for constructing the smooth dependence of \(\varepsilon - 1\) on \(h\), it was necessary to restrict oneself to determining mean values for each pair of neighboring measurements. This led to the calculated diagrams for

\[ \frac{\Delta(\varepsilon - 1)}{\Delta h} \]

having the form of figures composed of separate rectangles (“blocks”).

In Fig. 10 one of such diagrams is presented. On the right is given the curve of radio-echo intensity from various heights (\(\lambda = 123\) m). Freind indicates that in the case of sharp inhomogeneities this dependence (for \(\frac{d\varepsilon}{dh}=\text{const.}\)) would have the form of a continuously decreasing “wedge,” caused by uniform diffuse reflection from different heights. The presence of inhomogeneities creates projections and depressions on this “general background”—

\(^{1}\) This formula, proposed by Mumford\(^{35}\), is easily obtained from formulas (18) and (19) of our article\(^{4}\). We also note that the influence of water in the liquid and solid phases is in no way taken into account in this formula.

reflection from the troposphere, which is superposed on the background of fluctuations in the receiving apparatus. In the altitude interval from 0 to 4 km there usually exist several regions of sharp inhomogeneities, for which the reflection coefficient may reach \(10^{-4}\) (the reflection coefficient for the basic background is of the order of \(10^{-6}\)—\(10^{-7}\)). In general, the calculated and experimental data agree rather well; those cases in which this agreement is absent may be attributed to the fine structure of the troposphere, detected only by means of radio-echo observations.

1. Thomas and Colwell\({}^{36}\) approximately calculated the reflection coefficient \(\varphi\) for the case of normal incidence of a wave on a thin plane layer of thickness \(D\), placed between two media with different dielectric coefficients \(\varepsilon_1\) (below) and \(\varepsilon_2\) (above).

Fig. 10. Example of comparison of calculated values of \(\dfrac{\Delta(\varepsilon-1)}{\Delta h}\) with the intensity of the radio echo. Wavelength 123 m.

●—temperature, ○—elasticity of water vapor (Fraima)

The merit of their method is that they use not a “ray,” but a wave treatment, proceeding from the wave equation simplified as applicable to the case under consideration

\[ \frac{d^2 E}{dz^2} + k^2 \varepsilon(z)\cdot E = 0, \]

where

\[ k=\frac{\omega}{c} \]

is the wave number.

The dielectric coefficient of the considered (thin) intermediate layer is assumed by them to vary according to the law

\[ \varepsilon(z)=\varepsilon_1\left(\frac{z}{z_1}\right)^{2p} =\varepsilon_2\left(\frac{z}{z_2}\right)^{2p} \]

and they introduce the notation

\[ n=\left|\frac{z_1}{z_2}\right|^{p}; \]

here \(z_1\) and \(z_2\) are the ordinates of the lower and upper boundaries of the layer, and \(p\) is an undetermined exponent, about which it is known that \(p<1\) when \(n>1\).

For the field inside the layer one may approximately put:

\[ E=A\exp(j\Lambda+\zeta)+B\exp(-j\Lambda-\zeta), \]

where

\[ \Lambda=\frac{k\sqrt{\varepsilon_1}\,z_1^{p}}{(p+1)z_1^{p}} \quad \text{and} \quad \zeta=-\frac{p}{2}\ln z_1. \]

For the field at points located below the layer \((z<z_1)\), evidently,

\[ E_1=A_1\exp(jk\sqrt{\varepsilon_1}z)+B_1\exp(-jk\sqrt{\varepsilon_1}z). \]

An analogous expression is also obtained for the field at points located above the layer \((z>z_2)\). Substitution of these expressions into the boundary conditions leads to the formula for the reflection coefficient

\[ \Phi= \frac{ p\left(n^{1/p}-1\right)\sin\left[ k\sqrt{\varepsilon_1}D\left(\frac{n+p}{n+1}\right) \right] }{ 2k\sqrt{\varepsilon_1}D }. \tag{9} \]

It follows from this formula that the reflection coefficient can pass through zero, the “first zero” occurring for a layer thickness \(D_0\) determined from the condition

\[ kD_0=\frac{\pi}{\sqrt{\varepsilon_1}}\frac{n+1}{n+p}. \]

For the case of a very thin layer\({}^{37}\), characterized by a very small value of \(\Delta \varepsilon=\varepsilon_2-\varepsilon_1\), the exponent \(p\to 1\), and expression (9) may be approximately represented as

\[ \Phi=\frac{n-1}{2}. \]

But since \(\varepsilon_1\) differs only slightly from unity, \(n=\sqrt{1+\Delta\varepsilon}\simeq 1+\frac{\Delta\varepsilon}{2}\), and, consequently, \(\Phi=\frac{\Delta\varepsilon}{4}\).

It is obvious that this result corresponds to expression (7) for the reflection coefficient from a sharply bounded layer.

A somewhat different expression for the reflection coefficient from a diffuse layer was obtained by Studeley\({}^{38}\). Basing himself on the works of Darwin\({}^{39}\) and Har-

... he obtained the expression

\[ \Phi=\left|\frac{1}{4}\left(\sec^{2}\varphi-A\right) \int_{0}^{\infty}\frac{d\varepsilon}{dz}\, \exp\left\{-j\frac{4\pi\cos\varphi}{\lambda}z\right\}\,dz\right|. \tag{10} \]

where \(\varphi\) is the angle of incidence, and \(A\) is a certain constant equal to zero for horizontal polarization and to two for vertical polarization.

For normal incidence this expression coincides with the results of Försterling4. It is interesting to note that the relation between the reflection coefficients corresponding to the cases of normal and oblique incidence has the form

\[ \frac{\Phi(\lambda,\varphi)} {\Phi\left(\frac{\lambda}{\cos\varphi},0\right)} =\sec^{2}\varphi-1, \]

independent of the law of variation of \(\varepsilon\) with height.

For vertical reflection from tropospheric layers of waves of length 160 m, Studley obtained values of \(\Phi\) lying within the limits from \(10^{-7}\) to \(10^{-5}\).

1. Finally, there is also possible a conception in which the reflecting layers of the troposphere are regarded as an aggregate of small cloud-like inhomogeneities of the air, which may have the form either of flat formations or of globules.

Piddington[^18] made an attempt to compare the magnitudes of the reflection coefficients from such a globule, on the one hand, and from an infinite plane layer, on the other.

Fig. 11. Schematic diagram of a pulse transmitter (Harang and Shogfregen)

If, to simplify the conditions, it is assumed that the globule is bounded by a circular plane part facing the receiver, then the method of Fresnel diffraction is applicable. It then turns out that, for such a globule of diameter 540–775 m, located at a distance of 5 km, the reflection coefficient at \(\lambda=30\) m is twice as large as for a plane layer

with the same $\varepsilon$. If, however, one postulates that the side of the cloud facing the receiver is concave, then even larger values can be obtained for $\Phi$.

Fig. 12. Circuit diagram of the receiver (Harrang and Stoffregen)

Developing these ideas, it is probably possible to create a sufficiently plausible picture of reflection from layers made up of such globules, grouped at certain heights. The possibility of the existence of such layers (even if only sporadic ones) would easily explain the experimental results of Watson-Watt,^4 and possibly also the short periods of interruption of radio communication with the stratospheric balloon Explorer-II that occurred during its ascent.^42 Moreover, if one further assumes (this assumption is not devoid of probability) that effective reflection from such hypothetical layers can also be obtained when they are “illuminated” from above, then the possibility is not excluded of some distortion of the readings of aeronautical radio altimeters during flights over such layers.^1)

However, the geophysical probability of the formation in the troposphere of such strongly reflecting layers is not clear to us.

1. In conclusion of this section it is necessary to note the possibility of reflection of sounding pulses not only from tropospheric layers, but also from mountains, high buildings, and, finally, aircraft. Special investigations in this direction were carried out by Harrang and Stoffregen^20 in Tromsø. They worked at a wavelength $\lambda = 7.3\ \mathrm{m}$ (42 MHz). In Fig. 11 their transmitter circuit is given, operating into a half-wave vibrator with a reflector. With pulses of duration from $10^{-5}$ to $2 \cdot 10^{-5}$ sec, the power in the vibrator was of the order of 4 kW. The receiver circuit is shown in Fig. 12. The receiver had two stages of high-frequency amplification (from 30 to 60 MHz), a mixer, eight stages of amplification at the intermediate frequency (4 MHz), a detector, and two stages of low-frequency amplification. The selection of the circuit elements ensured normal reception of reflected signals from distances of 4–5 km.

With such apparatus, reflections were obtained from distances of the order of 8–20 km, which were quite stable in time. When the distance between the transmitter and receiver was increased

^1) A description of aeronautical radio altimeters, the operation of which is based on the reflection of radio waves from the earth, may be found in the articles by Magee^43 and Esenwein and Nieuwenhuyse.^44

to 150, 900, and 2,000 m the intensity of the reflected signals does not decrease. Further investigation showed that the intensity of the reflection depends strongly on the arrangement of the transmitting vibrator. In Fig. 13 are shown oscillograms corresponding to various experimental conditions: a—a horizontal vibrator with a reflector, placed along the east—west line; b—the same arrangement of the vibrator, but without a reflector; c—a horizontal vibrator without a reflector, placed along the north—south line; d—a vertical vibrator without a reflector.

For arrangements a and b, two reflections were obtained, from 8 and 14.8 km; for arrangement c—one reflection from 15.6 km; for arrangement d—a series of reflections from 8 to 15 km (curve e corresponds to the calibrating frequency of 3,000 MHz). Other experiments at a frequency of 11.5 MHz, with a power of about 50 kW in the pulse, gave a reflection from a distance of 17 km.

Fig. 13. Reflection from mountains (Tromsø, wavelength 7.3 m (Harang and Stoffregen))

Since Tromsø is surrounded by mountains 800–1,000 m high, located at distances of 5–20 km, the named authors explained the results they obtained by reflection from mountains, which can hardly be doubted. Indeed, the constancy of the reflections in time and their dependence on the orientation of the vibrators speak in favor of this.

It is obvious that in these experiments there are no tropospheric reflections of the kind reported by Watson Watt, Colwell, Englund, and others (see above). It is unlikely that this circumstance in any way compromises all experiments with reflection. Rather, one must suppose that under the conditions of the experiments at Tromsø the atmospheric conditions were not favorable for the creation of those sharp layers of inhomogeneity which are necessary for obtaining such reflections. That such circumstances evidently do occur may also be inferred from the results of Wainwright² in England, who likewise did not detect reflections (at any rate, regular ones) in his experiments—which, it is true, pursued quite different aims.

Let us also point out that Guin and Butcher⁹, comparing the results of the experiments of Colwell and Friend with the results of Watson Watt and others, expressed the supposition that the experimental conditions in America and England may be very different.

As regards the possibility of reflection of UHF from airplanes¹), a good illustration of such reflection for a wave of \(\lambda = 4.7\) m is given by Englund, Crawford, and Mumford¹⁰. In Fig. 14 are shown records of fields,

¹) The first mention of this question, see the article by Trevor and Carter⁵⁶.

of the field under the conditions of an aeronautical radio link. Partial changes of the field are due to the rapid change in the length of the path of the reflected rays (from the aircraft), interfering with the principal field at the receiving point.

Fig. 14. Field variations caused by reflection of short waves from an aircraft
(Englund et al.)

Some data on this question are contained in the papers by Rice^45b, Berrouza, Desinot and Hanta^45c, and in the review article by I. V. Brenev^45d. The possibility of such reflection is also mentioned by Appleton and Piddington^17.

§ 3. “TROPOSPHERIC WAVES” AND METHODS FOR CALCULATING THEM

1. Having postulated the existence in the troposphere of reflecting layers with relatively abrupt changes of \(K\), and their stability, one can determine the field strength produced at the receiving point as a result of reflection from these layers.

Fig. 15. Schematic course of the principal rays determining the field of a “tropospheric” wave

The first attempt to calculate such a field belongs to Englund, Crawford, and Mumford^30, who considered the case of equal heights of the transmitter and receiver. This case is shown schematically in Fig. 15. It is evident that the presence of a reflecting layer should lead to the arrival at the receiving point of at least four rays, the aggregate of which determines the field strength of the “tropospheric” wave.

We shall carry out all the reasoning, assuming the layer to be sharply bounded.

If the dielectric coefficient of the air between the earth and the reflecting layer did not depend on height, then refraction would be absent and all ray trajectories would be rectilinear. However, with increasing height \(e\) decreases, as a result of which refractive curvature of all trajectories occurs. The authors indicated postulate a linear decrease of \(e\)

with height (between the earth and the layer), and replace the true terrestrial radius \(a\) by the equivalent radius \(a_e = 1.33a\), after which the trajectories of all rays are taken to be rectilinear1).

Assuming that the reflecting layer of the troposphere is situated at an altitude of \(1500\ \text{m}\) and is characterized by the value \(\Delta\varepsilon = 10^{-7}\), Englund, Crawford, and Mumford calculated (for a radiated power \(P_\Sigma = 1\ \text{kW}\)) the dependence of the field strength of the tropospheric wave on distance for a transmitting-antenna height \(h = 42\ \text{m}\), a receiving-antenna height \(z = 5\ \text{m}\), and a wavelength \(\lambda = 4.7\ \text{m}\). In Fig. 16 two curves are given, corresponding to the cases of vertical \((A_V)\) and horizontal \((A_H)\) polarization. The calculation itself is not presented in the article. For comparison, diffraction curves for transmission over the sea \((\varepsilon = 80;\ \sigma = 5\cdot10^{-11}\ \mathrm{CGSM})\) are given in the same drawing. The curve for vertical polarization \((B_V)\) was calculated by Vvedensky’s formula[^47], and the curve for horizontal polarization \((B_H)\)—by Gray’s formula[^48].

Fig. 16. Theoretical curves of the dependence of the field strengths of tropospheric and diffraction waves on distance (Englund, Crawford, and Mumford).

In calculating these curves, Englund et al. introduced into the diffraction formulas the equivalent terrestrial radius \(a_e = 1.33a\). Such a replacement, in their opinion, makes it possible to take into account the influence of mean refraction on diffraction. However, as we indicated in our article1, the validity of such a procedure, which leads to a decrease in the slope of the rectilinear part of the diffraction curves (drawn on a logarithmic scale), is at least unproven2. At present, one of us has obtained a diffraction-refraction formula showing that the quantity \(a_e\) affects the course of the curves in an essentially different way[^1,^48]. Owing to this, the diffraction-refraction curves do not run above the diffraction curves without allowance for refraction, but below them (in the case under consideration by \(7\)–\(12\ \mathrm{db}\), depending on the distance), and this applies both to vertical and to horizontal polarizations.

1. From Fig. 16 it is clear that near the horizon, at distances up to the horizon for given \(h\) and \(z\) equal to \(36\) km (taking into account the correction for mean refraction), the diffraction field at \(30\)—\(40\) db exceeds the field of a tropospheric wave [or, with the indicated (*) correction, by \(20\)—\(30\) db]. In this zone, according to Englund et al., reception should be relatively stable\(^1\). At the receiving point (East Moriches), located at a distance of \(112\) km from the transmitter, the field strengths of tropospheric and diffracting waves are already quite comparable; this will remain true also after introducing the indicated correction. The interference of these fields, in the opinion of Englund et al., makes reception at such distances less stable.

Fig. 17. Theoretical dependence of the field of the tropospheric wave (in db relative to the free-space field) on the height of the reflecting layer. Distance \(112\) km, \(\lambda = 4.7\) m (Englund, Crawford, and Mumford)

Finally, at still greater distances the diffraction field becomes altogether insignificant, and reception is determined only by the tropospheric wave. Naturally, with the disappearance of the layer, such reception ceases. It is very possible that it is precisely this that explains the cases of relatively very strong, but also very inconstant, reception of meter waves at distances of the order of \(500\)—\(800\) km, described by Oxman and Plendl\(^ {50}\).

In Fig. 17 is given the dependence of the field of the tropospheric wave on the height of the reflecting layer and on the value of \(\Delta \varepsilon\), calculated by Englund et al. for reception at East Moriches (\(112\) km). Owing to the dependence of the reflection coefficient of the layer on the polarization, the influence of the height of the reflecting layers also depends on the polarization. For vertical polarization (\(h = 42\) m and \(z = 5\) m) the field decreases monotonically up to a layer height of \(4700\) m; at the same time, for horizontal polarization (\(h = 45\) m and \(z = 9.5\) m), at a height of \(3000\) m there is a deep minimum. However, for very low-lying layers the influence of polarization is insignificant.

Having experimental data (for 45 days) on the presence of reflecting layers, their height, and \(\Delta \varepsilon\), the authors cited compared the field-strength values calculated on the basis of theoretical considerations with those actually observed. The results of this comparison are given in Fig. 18.

The triangles show the ratios of the calculated fields of tropospheric waves to the “free-space” field (\(A\)-component). The ratios of the diffraction fields, calculated by Vvedensky’s formula (but with equi—

\(^1\) In expressing this judgment, Englund, Crawford, and Mumford apparently underestimated the influence of changes in the mean refraction on the diffraction field; see § 4.

equivalent earth radius \(a_e\)), to the free-space field (\(B\)-component) are shown by circles. The vertical lines correspond to the relative values of the observed fields. The upper ends of these lines give the maximum field values, and the lower ends give such “average” fields as are exceeded during 50% of the observation time and lie above the fields measured during the remaining 50% of the time.

During the period when no layers were observed, the field strengths calculated by the diffraction formulas with \(a_e\) exceeded the observations on the average by 8 db; nevertheless, the authors mentioned tend

Fig. 18. Comparison of theoretical and experimental data for radio transmission over 112 km. \(\lambda=4.7\) m; vertical polarization; \(h=42\) m; \(z=5\) m.

1 — clearly expressed layer with \(\Delta\varepsilon \simeq 10^{-3}\); 2 — no layer; 3 — weak layer with \(\Delta\varepsilon \simeq 10^{-3}\); 4 — indefinite layer; 5 — possible layer below 400 m; \(\Delta\) — calculated \(A\)-component; \(o\) — calculated \(B\)-component (Englund, Crawford, and Mumford)

to see in the comparison presented good confirmation of the validity of their calculations. However, in the light of the diffraction-refraction formula indicated above, the latter consideration finds a complete explanation. Indeed, for a distance of 112 km, using the new formula we find field values 10 db smaller than according to the formula applied by Englund et al., which agrees quite satisfactorily with the field values observed by them in the absence of layers.

3. In 1940 Norton\(^ {51}\) published, in the materials of the American Federal Communications Commission, a calculation of the field of tropospheric waves. Postulating a sharp boundary of the reflecting layer (Fig. 19), he made use of the usual expressions for the Fresnel reflection coefficients.

For horizontal polarization \(\vec E\), for this coefficient, as is known,\(^1\) we have:

\[ F=\frac{\cos\varphi-\sqrt{\varepsilon_1-\sin^2\varphi}} {\cos\varphi+\sqrt{\varepsilon_1-\sin^2\varphi}} . \]

\(^1\) See, for example, \({}^{31}\), § 2.1.

From this expression, in view of the extraordinary smallness of \(\Delta\varepsilon=\varepsilon_2-\varepsilon_1\approx \varepsilon_2-1\), we obtain approximately:

\[ F \approx -\frac{\Delta\varepsilon}{4\cos^2\varphi} \left(1-\frac{\Delta\varepsilon}{4\cos^2\varphi}-\cdots\right). \tag{11} \]

Similarly, for vertical polarization, for the reflection coefficient we have:

\[ f= \frac{\dfrac{\varepsilon_2}{\varepsilon_1}\cos\varphi -\sqrt{\dfrac{\varepsilon_2}{\varepsilon_1}-\sin^2\varphi}} {\dfrac{\varepsilon_2}{\varepsilon_1}\cos\varphi +\sqrt{\dfrac{\varepsilon_2}{\varepsilon_1}-\sin^2\varphi}}, \]

whence, approximately, we obtain:

\[ f\approx \frac{\Delta\varepsilon}{2} -\frac{\Delta\varepsilon}{4\cos^2\varphi} +\left(\frac{\Delta\varepsilon}{4\cos^2\varphi}\right)^2 . \tag{12} \]

If refraction in the atmosphere on the path to the reflect-

Fig. 19. Notation; see formula 13

ing layer is neglected, then the angle of incidence \(\varphi\) (Fig. 18) is determined as follows:

\[ \tg\varphi= \frac{a\sin\vartheta}{a(1-\cos\vartheta)+H}, \]

which, for relatively small \(R<2R_1\), leads to the formula:

\[ \cos^2\varphi= \frac{\left(\dfrac{R^2}{8a^2}+\dfrac{H}{a}\right)^2} {\dfrac{R^2}{4a^2}\left(1+\dfrac{H}{a}\right)+\left(\dfrac{H}{a}\right)^2}. \tag{13} \]

In Fig. 20 are given the values of the reflection coefficients, calculated by Norton from formulas (11) and (13), for layers at heights 0.75, 1.5, and 3 km for three values of \(\Delta\varepsilon\): \(10^{-4}\), \(10^{-5}\), and \(10^{-6}\).

From Fig. 18 it follows that \(\varphi\) has its greatest value when \(R=2R_t\), where \(R_t=\sqrt{2aH}\) is the distance to the horizon for a point (of reflection) situated at height \(H\). In this case

\[ \cos^2 \varphi_{\max}=\frac{4H}{2a+3H}, \]

which also determines the region in which an approximate “reflection” interpretation of the phenomena under consideration is possible. Norton, however, uses this value \(\varphi_{\max}\) also at distances exceeding \(R_0\), attempting in this way to connect diffraction propagation around the earth with reflection from a layer.

Obviously, a rigorous solution of this diffraction problem would require a mathematical treatment in the spirit of Watson’s well-known work\({}^{52}\), devoted to the diffraction of radio waves in the space between the earth’s surface and the concentric ionized layer surrounding it.

Fig. 20. Reflection coefficients from the layer and interference multipliers. Horizontal polarization (Norton).

4. Having restricted himself to considering rays undergoing a single reflection from the layer, Norton combined the points of this reflection into one.\({}^{1}\) In radio transmission within the horizon (with respect to the point of reflection from the layer), the approximate formula for the field intensity (amplitude value) may be represented in the form

\[ E=\frac{300\sqrt{P_\Sigma}}{R}\,F\cdot 2\sin\left(\frac{4\pi h'H'}{\lambda R}\right)\cdot 2\sin\left(\frac{4\pi z'H''}{\lambda R}\right). \tag{14} \]

The first factor of this formula determines the field of a unit elementary dipole (“free-space field”), decreasing inversely proportional to the distance; the reflection coefficient \(F\) takes into account the attenuation of the field upon reflection from the layer (horizontal polarization). The remaining two factors take into account the interference structure of the field (“interference multipliers”). Here \(h'\), \(H'\), \(z'\), and \(H''\) are the so-called

\({}^{1}\) Using the example of a plane earth and a plane reflecting layer (see Appendix 1), it is easy to verify that when all points of reflection of these rays are transferred into one, the phase relations are practically not disturbed. In the real case of a spherical earth, this combination is less rigorous. However, it hardly makes sense to take this circumstance into account in detail, in view of the obviously incorrect form of the reflecting layer and the neglect of all other rays.

“reduced heights,” determined by the approximate formulas\(^1\):

\[ h' = h - \frac{R^2}{2a_e}\left(\frac{h}{h+H}\right)^2 \quad \text{and} \quad H' = H - \frac{R^2}{2a_e}\left(\frac{H}{H+h}\right)^2 \]

and, analogously,

\[ z' = z - \frac{R^2}{2a_e}\left(\frac{z}{z+H}\right)^2 \quad \text{and} \quad H' = H - \frac{R^2}{2a_e}\left(\frac{H}{H+z}\right)^2 . \]

In Fig. 20 are shown the dependences of the interference multipliers \(\xi(h,H,R)=A_1\) and \(\xi(z,H,R)=A_2\) on the distances \(R_1\) and \(R_2\), equal to \(0.5R\), computed by Norton for a frequency of 50 MHz at heights \(h=305\ \text{m}\), \(z=9\ \text{m}\), and \(H=1.5\ \text{km}\). The difference between these curves is explained by the fact that \(h>z\), as a result of which the path differences between the interfering rays (the direct rays and those reflected from the earth) are different. In Fig. 21 the dependence of the field of the tropospheric wave on the distance \(R=R_1+R_2\) is given, computed by Norton from formula (14) for the case of horizontal polarization with \(h=305\ \text{m}\) and \(z=9\ \text{m}\). The value \(\Delta\varepsilon\) was taken equal to \(10^{-5}\), and \(H=1.5\ \text{km}\). For comparison, the same figure also gives the diffraction curve computed by Norton for transmission over land \((\varepsilon=15\) and \(\sigma=5\cdot10^{-14}\ \mathrm{CGSM})\).

Fig. 21. Theoretical dependence of the field of the tropospheric wave on distance, \(\lambda=6\ \text{m}\); \(h=305\ \text{m}\); \(z=9\ \text{m}\); \(H=1.5\ \text{km}\); \(\Delta\varepsilon=10^{-5}\); \(P_{\Sigma}=1\ \text{kW}\) (Norton)

Norton considers the influence of changes in \(\Delta\varepsilon\) within the limits from \(10^{-4}\) to \(10^{-6}\); of the layer height \(H\) within the limits from 0.75 to 3.0 km; of the transmitter height \(h\) within the limits from 30 to 300 m; and, finally, of the frequency within the limits from 50 to 300 MHz. From the curves he gives it follows that an increase in \(\Delta\varepsilon\) increases the field of the tropospheric wave; an increase in \(H\) decreases this field at small \(R\) and increases it at large \(R\); an increase in frequency also increases the field (within certain ranges of distance). These curves show that, under meteorologically possible variations of \(\Delta\varepsilon\) and \(H\), the field of tropospheric waves may change by hundreds of times. Therefore, despite the general geophysical interest of all these calculations, their practical value is still very problematic.

The question arises of the need to take into account the influence of all the remaining rays that undergo multiple reflection from the layer and from the earth. An approximate accounting of the totality of all these rays shows that, owing to the small value of the reflection coefficient from the layer, one may

\(^1\) See, for example, \({}^{31}\), § 6.1, sec. 2. The question of a more exact determination of the reduced heights was considered by Arenberg and Pekeris \({}^{31a}\).

to confine ourselves merely to taking account of rays reflected once from the layer. In view of this, formula (14) may be regarded as quite a sufficient approximation (see Appendix 1). At the same time it is very probable that, by carrying out a sufficiently consistent analysis for the whole set of rays, one may arrive at results close to those which a rigorous theory—perhaps based on the above-mentioned work of Watson—should give.

Thus, the calculations presented are for the time being capable only of giving an idea of the magnitude of the fluctuations of the observed fields due to tropospheric reflections. It is possible that subsequently a special “troposphere service” will be organized, analogous to the “ionosphere service” now practiced rather widely, and more purposeful than the existing meteorological service. Perhaps then such calculations will acquire a much firmer foundation.

We shall also point out that if we were dealing not with a specularly reflecting layer, but with a separate globule situated at a height \(H\), then the field at such a globule from a transmitter located within its horizon would be approximately equal to

\[ E_1 \simeq \frac{300\sqrt{P_\Sigma}}{R_1}\cdot 2\sin\left(\frac{2\pi h'H'}{\lambda R_1}\right). \]

Fig. 22. Schematic path of rays upon reflection from a globule

Further, considering this globule as a certain fictitious radiator, for the field produced by it at the receiving point (also located within the horizon), we would have the expression

\[ \begin{aligned} E &\simeq \frac{KE_1}{R_2}\,2\sin\left(\frac{2\pi z'H''}{\lambda R_2}\right) == \\ &= K\frac{300\sqrt{P_\Sigma}}{R_1R_2}\,2\sin\left(\frac{2\pi h'H'}{\lambda R_1}\right)\cdot 2\sin\left(\frac{2\pi z'H''}{\lambda R_2}\right). \end{aligned} \tag{15} \]

Here \(K\) is a certain diffraction coefficient, taking into account the character of the “secondary radiation” of the globule; \(R_1\) and \(R_2\) are distances, in the present case not necessarily equal to one another (Fig. 22).

Comparing this formula with formula (14), we see that in the case of a globule there is a factor \(\dfrac{K}{R_1R_2}\), where \(R_1 \ne R_2\), which in the case of a specularly reflecting layer passes into the factor \(\dfrac{f}{R_1+R_2}\), where \(R_1=R_2\). A collection of individual globules may give an even more complicated picture of the field. Above we cited some considerations of Piddington\(^{18}\) relating to this question (see § 2, item 5).

1. All the preceding arguments concerning tropospheric waves postulated the absence of dispersion. It is true that, according to Holmes\(^{53}\), in moist air, under the action of a radio wave, oscillations of water molecules are possible, the dipole moment of which experiences, in the electric field of the Earth, the action of a directing pair of forces. This introduces into the consideration

dispersion of radio waves, which, of course, greatly complicates all theoretical computations. However, as V. L. Ginzburg shows[^53a], Holmes’s assumptions do not correspond to reality.

§ 4. FIELD VARIATIONS IN UHF RECEPTION

1. For practical applications, the issue of cardinal importance is the assessment of the stability of UHF reception and the dependence of this stability on the location of the corresponding points, the terrain relief and the nature of the soil, the wavelength and its polarization, the time of day and season, and, finally, meteorological conditions. All these factors, of course, interact, and separating their influence is inevitably associated with a very considerable degree of uncertainty. Nevertheless, such separation is naturally indispensable.

Experimental work on the propagation of UHF up to 1935–1936 contained almost no quantitative data concerning the constancy of reception strength under various conditions, and provided only a very vague qualitative assessment1. Work of recent years, in which modern apparatus was employed, has yielded very considerable quantitative material, though relating chiefly to the propagation of waves only in the meter range.

The greater part of these works belongs to American authors; therefore, naturally, all these investigations relate to the climatic conditions of America (mainly to the districts in the vicinity of New York). There is incomparably less information in the literature concerning investigations in other countries; it is exhausted by data from a study of the reception of television transmissions near London and, apparently, by specially organized investigations on the propagation of meter waves in Germany.

Questions of reception stability on decimeter waves have been investigated very incompletely. In this range there are only a few scant quantitative data relating to conditions in the USA and the English Channel.

1. The mutual location of the corresponding points, of course, cannot fail to influence the stability of communication on UHF, for the relation between the ground and tropospheric components of the field at a given receiving point depends strongly on this location. For the same reason, the nature of the terrain relief and soil between the corresponding points is important.

As early as 1935, Burrows, Decino, and Hunt[^61], experimenting on a wave of 8.65 m, noted a progressive increase in fadings with increasing distance2. This proposition is fully confirmed by the results of the later work of McLean and Wickizer[^62]. Their transmitting antenna (horizontal vibrators) was installed on the Empire State

Building at a height of about 396 m above the ground. Observations at frequencies of 52 MHz and 49.5 MHz were conducted simultaneously at three points located at distances of 51, 115, and 275 km. The first of these points was within the transmitter’s horizon.

Table 3

Field variations during reception at 52 MHz

Location	Day, db min	Day, db max	Night, db min	Night, db max	Largest variations during the day
New Brunswick (51 km)	0.5	2	0.5	3	10
Momlegen (115 km)	2	17	2	30	25
Roystertown (275 km)	15	40+	3	35--	49+

Table 3 gives the values of the maximum and minimum changes of the field (in decibels) that occurred during 30-minute daytime and nighttime observations at the indicated points (June–July 1937). The last column of the table gives the maximum field variations during the day.

This table confirms the increase in the depth of field variations with distance. Obviously, at small distances such a dependence is caused mainly by refraction of ultra-short waves in the very lowest layers of the troposphere; as the distance increases, the change in the relative magnitude of the ground and tropospheric components of the field begins to play a role. This can be verified on the basis of the experiments of Englund, Crawford, and Mumford,^3 who carried out their observations near New York at distances of 62.7 km (reception on a hill) and 65.9 km (reception behind a hill). For transmission on a wavelength of 4 m the transmitting antenna was raised to a height of 9.15 m, the radiated power was about 0.5 W; when operating on a wavelength of 2 m the height of the transmitting antenna was 4.33 m, the radiated power about 2–4 W; the polarization was horizontal.

As can be seen from the field records (Fig. 2), already cited by us earlier, reception behind the hill was accompanied by deeper field variations, which should be attributed to the weakening of the ground component of the field.

1. As regards the influence of the height to which the antenna is raised, then at heights ensuring the possible maximum of the ground component of the field, reception should possess the greatest stability. This stability must be achieved not only by a relatively large, in comparison with the tropospheric component of the field, ground component, but also by the fact that at such “optimal” heights the variations of the coefficients characterizing the mean refraction of ultra-short waves in the troposphere should have a minimal influence.

On the basis of this proposition, Kroger, Trevor, and Smith^63, as a measure for combating fading, recommended the placement of transmitting and receiving antennas so as to provide a path difference between the direct and reflected (from the Earth’s surface) rays equal to

\[ \Delta r = \frac{\lambda}{2}. \]

However, as

elementary calculations show (see Appendix II), practical fulfillment of this condition would require the construction of very high masts; therefore Kroger et al. recommend limiting oneself to smaller heights, for example those providing a path difference \(\Delta r=\frac{\lambda}{6}\). Taking into account the phase change of the field upon reflection from the earth, such a value of \(\Delta r\) corresponds to a phase shift between the direct and reflected rays of \(120^\circ\). In this case the field at the receiving point is numerically equal to the field of a half radiator. However, there is no need to prove specially that this equality cannot be interpreted as an exclusion of the influence of the Earth, and in no way as an exclusion of the tropospheric wave. Therefore the value recommended by the cited authors, \(\Delta r=\frac{\lambda}{6}\), does not correspond to any special conditions of propagation of ultrashort waves. It is obvious that for values \(\Delta r>\frac{\lambda}{6}\) the field will be less sensitive to variations of \(\Delta r\), and for \(\Delta r<\frac{\lambda}{6}\) more sensitive.

The indications of Ferson and Ulrich\({}^{64}\) may also be included here; they noted that transmission across the English Channel on a wave of \(6\ m\) was considerably more stable than transmission on a wave of \(17\ cm\). This can be explained by the fact that the phase difference between the direct and reflected rays, expressed as \(\frac{2\pi\Delta r}{\lambda}\), is obviously much more sensitive to changes in the refractive coefficients at a shorter wave than at a longer one. Hence one is led to the conclusion that the field is relatively more stable in transmissions on longer waves. However, the recent experiments of Englund et al.\({}^{3}\) show that such a conclusion is not always valid. Indeed, from their observations of the stability of the field in transmissions on waves of 4, 7, 4 and \(2\ m\), the indicated conclusion cannot in any way be drawn (see Fig. 2).

Fig. 23. Field records during simultaneous reception by closely spaced receivers. Distance from the transmitter \(90\ km\) (Scholz and Eggersdörfer)

1. A very interesting question is the simultaneity of variations of fields observed during reception at different points. Scholz and Eggersdörfer\({}^{27}\), who specially investigated this question, indicate that, with a separation of the receiving stations of \(14\ km\), coincidences in the form of the field records of two closely spaced transmitters operating on waves \(7.5\) and \(7.18\ m\), during reception at a distance of \(200\ km\), as a rule were not observed. Good agreement was obtained only with a considerably closer spacing of the receivers. Fig. 23 gives an idea of the character of such records,

received when receiving waves of 6.77 and 7.06 m at a distance of 90 km (Berlin—Windsleben).

Valuable indications on this question are given by Weinick,² who studied the propagation of vertically polarized waves of 6.66 and 7.23 m from the transmitters of the London television center at Alexandra Palace.¹ The receiving point was 900 m below the horizon line. In his observations Weinick used two separate receivers with vertical half-wave vibrators, raised to a height of \(\frac{\lambda}{2}\) above the ground. The voltages from these receivers were fed to a common cathode oscillograph.

In Fig. 24 are given field recordings obtained with receiver spacings in a direction perpendicular to the line of communication, of 3, 27, and 100 wavelengths.

Fig. 24. Field recordings when receiving on two separate, spaced receivers. Spacing:
\(a\)—by \(3\lambda\); \(b\)—by \(27\lambda\); \(c\)—by \(100\lambda\). (Weinick)

As we see, with a spacing of \(3\lambda\) both the rapid and the constant variations of the field coincide completely; with a larger spacing the rapid fading no longer coincides. With a further increase in the distance between the receivers, the coincidence of the slow fading also ceases. Obviously, by developing such experiments, one can estimate the degree of distortion of the wave front.

As for the influence of polarization, here, according to the available theoretical and experimental data, one must clearly distinguish the case of propagation of ultrashort waves over the sea from the case of propagation over land. This is explained by the fact that, in the propagation of ultrashort waves over seawater, at comparatively small heights the ground component of the field, for vertical polarization, is considerably greater than for horizontal—

¹ A detailed description of this center may be found in the article by McNamara and Birkinshaw,⁶⁵ in which the results are given of measurements of the field strength in London and its environs. Some data on this question are also given in ³¹, p. 129.

field. In propagation of u.s.w. over land, the field component depends comparatively little on the polarization used.

As an example we shall cite field records (Fig. 25) obtained by Englund et al.^30 in transmission over the sea over a distance of 112 km. The transmission

Fig. 25. Field records in transmission over the sea. Different polarizations (Englund, Crawford, and Mumford)

at a wavelength of 4.74 m was carried out with two polarizations simultaneously. The transmitting antennas were on the seashore at a height of 42 m; the receiving antennas were located almost at the water’s edge at a height of 5 m. These authors worked with rhombic antennas, V-shaped and half-wave. As can be seen, the depth of fading for horizontal polarization is considerably greater than for vertical polarization.

Fig. 2. Field records in transmission over land. Different polarizations (Englund, Crawford, and Mumford)

In Fig. 26 we give the results of field records obtained by Englund et al. with different polarizations in transmissions over land at a wavelength of 2 m over a distance of 65.9 km. From these curves it follows that no noticeable difference is observed in the slow variations of the field in transmission over land. However, the rapid variations of the field can be very different.

1. A quantitative comparison of the results obtained by different authors appears very difficult not only because of the difference in distances, but also because of the difference in the character of the terrain. Moreover, the picture is made still more confused by the random choice of the time of experimentation, its total duration, and the difference in methods of processing.

The first systematic observations of the stability of the field on meter waves apparently date to 1935. R. L. Hall²⁶, to whom many authors later referred, then published several curves characterizing field variations during radio transmission on waves of about 5 m, over distances on the order of 175 km.

In his later work⁶⁶ he gives a description of the apparatus, from which one should note the integrator for analyzing field records.

In Fig. 27 a general view is given of an installation consisting of a receiver, a recording instrument, and a group of electric clocks switched on by means of relays. For the alternate switching-on of these clocks there was a special device made of a rail with contacts, fixed to the frame of the recording instrument. The contacts were arranged so that the transfer of the closing device from one to another corresponded to a displacement by 5 db of the pen of the recorder registering the signal level. At first the closing of these contacts was carried out manually by the experimenter himself, who closed the contact near which the recorder pen was located. Later this device was automated. Comparison of the readings of the individual clocks with those of the main continuously running clocks made it possible to determine the percentage of time during which the signals had one level or another. Replacement of the clock faces by scales with 100 divisions made it possible to read the time corresponding to a given signal level directly as a percentage of the total observation time. Some data on similar devices may also be found in the article by Groskopf and Focht⁶⁷.

Fig. 27. General view of the recording device with a signal-level analyzer (R. Hall)

1. A major aid in placing the question of systematizing observations of the stability of ultra-short-wave reception on a technical basis is the introduction of time characteristics of reception stability. An example of such a characteristic, compiled by Englund et al.³, we saw in Fig. 3. This characteristic was constructed on the basis of processing observations of field variations over the whole of 1938, for reception of a 2 m wave at a distance of 65.9 km. Along the ordinate axis is plotted the time during which the signal was above its calculated value; the latter in this case was equal to the free-space field (0 db).

It follows from this curve that the signal was less than the free-space field for only 13% of the total time; moreover, it was lower by more than 5 db for only 0.1% of the total time. Such weak fields almost always occurred at night and in the summer months (see below). Signals lower by 10 db were observed for only about 99.5% of the total time; signals of an especially high level, reaching an excess of 13 db, were observed only for 0.1% of the total time.

Curves characterizing the time during which the field variations have one range or another are also given by Barrow, Desin, and Hunt[^45], who experimented at a wavelength of 2 m over a distance of about 60 km. The transmitting and receiving antenna devices consisted of four and six horizontal half-wave vibrators. The transmitting antenna was raised to a height of 70 m above ground level (143 m above sea level); the heights of the receiving antennas were 24 m and 50 m above ground level (70 m and 73 m above sea level).

Fig. 28. Percentage of time during which the field variations (per hour) are less than the indicated values (Barrow, Desin, and Hunt)

The stability of the signals when received at the end of communication lines may be judged from Fig. 28, which indicates the percentage of time during which the field variations over 1 hour were less than the stated values. For example, fields whose variations over 1 hour were less than 1 db were observed during 40% of the total time, etc.; here the mean annual field for 1936 was taken as the zero level. Barrow, Desin, and Hunt also give a curve that agrees well with the curve of Englund et al. reproduced by us in Fig. 3.

A comparison of individual curves obtained by different authors in America gives, in general, fair agreement. However, the practice of constructing such curves is still, unfortunately, confined only to the USA. This circumstance does not make it possible to form a clear judgment regarding the influence of climatic conditions on reception stability at ultra-short waves, and consequently regarding the possibility of transferring the experience of one country (for example, the USA) to the conditions of another.

As for the decimeter-wave range, very little literary data are known concerning the stability of the field when they are received. Some quantitative material characterizing this stability is contained in the article by Fershond and Ulrich[^64]. These authors experimented at a wavelength of 19.4 cm (see[^31], pp. 50 and 270) over a distance of 56 km (Lymn—St. Inglevert). However, it is very difficult to draw any sufficiently clear quantitative conclusions on the basis of their work.

There are also statements by Kroger, Trevor, and Smith[^65] concerning the stability of the field when receiving waves of 60 cm at a distance of 48 km. These authors indicate that, when receiving within the horizon, “fading of more than 10 db occurred rarely and then only for short periods, less than an hour.” As we see, all this information is extremely meager, and the question of the stability of reception of decimeter waves requires serious systematic study.

1. All American authors unanimously note the existence of a distinct seasonal and diurnal course of fading, indicating that the main—

...the course of this change does not depend either on the length of the wave or on polarization. In Fig. 29 a summary is given of two years of observations by Englund et al., showing a very clear picture of the character of the seasonal variation.

The diurnal variation is illustrated rather clearly by another diagram, presented in Fig. 30. From this diagram it follows that the most stable reception is observed at midday; sometimes such reception lasts for several (4–6) hours. At night, as a rule, intense fluctuations of the field are observed. However, this diurnal variation depends strongly on the time of year. The diurnal variation is most pronounced in summer; in the winter months the diurnal variation is expressed less clearly.

Fig. 29. Dependence of the depth of field variations on the time of year (Englund, Crawford and Mumford).

To this is added the further fact that the mean value of the field also has a diurnal and seasonal variation, as indicated by both American and English authors. Thus, Joua^54, Colster^59, Trevor and George^60, R. Hell^26, Burrows et al.^61 note that the mean daytime field is weaker than the nighttime field. Similarly, the mean signal level in winter is stronger than in summer.

Fig. 30. Diurnal variation of field variations in the winter and summer months (Englund, Crawford and Mumford).

§. A very important factor, playing a particularly important role in assessing the possibility of direction finding on meter and decimeter waves, is the question of the constancy of the direction of arrival of the waves at the receiving point. Recently this question has gradually begun to be covered in the literature.

Thus, Ferson and Ulrich^64 consider possible a deviation of decimeter waves, to which they attributed part of the field variations they observed. However, the data they cite are so unconvincing that they themselves are not certain of them.

More definite statements on this subject belong to Waynick^2, who specially studied this question over the course of two months. From the results he gives it follows that, in reception...

waves of 7.3 m at distances of the order of 70 km, the deviation of the received ray may have extremely large values, reaching up to ±11.5°.

These figures are very surprising, and in general this very important question has as yet been very little studied and demands the closest attention.

1. The correlation of the propagation of ultrashort waves with meteorological factors began to be sought from the very moment when the idea of refraction of ultrashort waves in the troposphere began to take hold. Fragmentary information on this question goes back as far as Joua,⁵⁴ who noted that on hot sunny days reception on waves of 5–6 m (Nice—Corsica line, \(R = 205\) km) was weaker than in cloudy weather. Indications of correlation with the weather may also be found in Marconi’s investigations⁵⁵ on the propagation of decimeter waves (\(\lambda = 57\) cm, \(R = 175\)–270 km). In the experiments of the VEI expedition³² on the Black Sea (\(\lambda = 60\) cm, \(R = 40\)–100 km), a correlation was observed between strong reception and the presence of refractive visibility of the Pontic mountains lying beyond the horizon. Scilling et al.⁶⁸, Englund et al.⁶⁹, Kolster⁵⁹, Trevor and George⁶⁰ also noted a connection between reception strength and meteorological factors in the broad sense of the word (diurnal and seasonal variations in the strength of ultrashort-wave reception).

Fig. 31. Reflection of radio pulses (\(\lambda = 125\) m) from the troposphere during the movement of air masses (Friend and Colwell)

R. Hall²⁶ was apparently the first to note the role of temperature inversions in the propagation of ultrashort waves in the troposphere and pointed out the importance of determining the vertical temperature gradient. His articles contain definite indications of a correlation between storms and strong signals; he also pointed to the effect of the intrusion of masses of tropical air, the consequence of which is usually an increase in temperature at heights of the order of 2000–3000 m, with a simultaneous strengthening of the signals. Subsequently these considerations found confirmation in the work of Friend and Colwell.¹⁶ There a special diagram is given (Fig. 31), illustrating the connection between the height of reflection of 125-m waves from the troposphere and the movement of air masses.

Fig. 32, from a later article by Friend,⁷⁰ gives the change in the height of the reflecting layer in South Lexington (U.S.A., Massachusetts) during a tropical storm (the night of 5 to 6 November 1939). The center of the storm was characterized by stable reflection. During the passage of the cold front the height of the reflecting layer decreased sharply. This is in good agreement with the curves of the distribution of temperature and humidity of the air with height, obtained during this storm. In the discussion concerning Friend’s article, Diamond⁷¹ expressed a number of considerations about

reflection coefficient. The order of the values obtained by him is approximately the same as in other authors.

Attempts to establish a correlation with the weather were also made by Ferson and Ulrich^64 ($\lambda = 17$ cm, $R = 56$ km). However, it is very difficult to extract any clear results from the data they present (see ^31, p. 270). Among their conclusions one should note the assertion that stable reception can occur in two cases: 1) when the air, being in a state of turbulent motion, does not contain inhomogeneities of a layered character and “pockets” that differ strongly in temperature and humidity, and 2) in the case of static equilibrium of the air.

Fig. 32. Reflection of radio waves during a tropical storm (Freund)

More complete data on this question began to appear only from the moment when researchers started to combine the study of UHF propagation with simultaneous observations of the state of the troposphere. As one of the first works in this direction one should note the work of Englund et al.^72, in which they noted a connection between the strength of UHF reception and the humidity of the air (see ^31, pp. 263 and 1).

A number of items of information may be gleaned from the works of Burrows et al.^45, Englund et al.^30, Scholl and Egertscheffer^27, Wainika^2, and Colwell, who, in cooperation with Freund, Bowden, and Thomas^8, ^11, ^12, ^15, ^33, ^36, ^37, devoted his activity to the study of the influence of meteorological factors on the propagation of radio waves of all ranges. Of greatest interest are the works of Freund and Colwell^16, Freund^34, ^70, and Englund et al.^3. Some information can also be obtained from Ferrell^73 and Nuaze^74. In the USSR, Nasilov^75 and Barakan^76 are engaged in these questions.

Summarizing all these data, it may be considered that clear, calm weather favors the appearance of fadings; wind and rain reduce fadings. However, there are often exceptions to this rule. Thunderstorms and storms occurring along the path of UHF propagation, as a rule, weaken fadings. Complete sporadic disappearance of reception is observed on quiet, clear nights. It has not yet been possible to establish more precise regularities and to cast them in quantitative form. Obviously, in order to establish such regularities, further accumulation of experimental data and their systematization are necessary.

CONCLUSION

All that has been set forth makes it possible to form a fairly coherent conception of the propagation of UHF in the troposphere. However, these considerations can give only a qualitative explanation of the regularities that are becoming apparent; for quantitative judgment, necessary for the rational design of radio links, these data are still insufficient.

One of the most obscure points complicating design is the question of the influence of climatic conditions, which has as yet been studied very little. As a shortcoming still inherent in almost all the corresponding works, it should be noted that meteorological measurements are usually made on the surface of the Earth at one definite point, somewhere along the communication line or near one of the corresponding points, and only occasionally at different altitudes by means of an airplane. Therefore, for communication lines of relatively great length, meteorological data prove insufficient for establishing clear quantitative relationships.

It must be supposed that the systematic development of these works will make it possible to establish a sufficiently clear correlation between the propagation of ultra-short waves and meteorological factors. This may provide the possibility of using data obtained from an analysis of the propagation of ultra-short waves in compiling weather forecasts. For the present there are only separate, rather vague indications on this matter. The decisive role in studying this range of questions belongs to mass experiment.

Appendix I

Let us consider the case of a flat earth (Fig. 33). Let the transmitter be placed at point \(A\). Then for the field at point \(B\), produced by rays with a single reflection from the layer, we have:

\[ E_1 = 300\sqrt{P_{\Sigma}}\cdot F_1\left(\frac{e^{jkr_1}}{r_1}+\frac{e^{jkr_2}}{r_2}-\frac{e^{jkr_3}}{r_3}-\frac{e^{jkr_4}}{r_4}\right)e^{-j\omega t}. \]

In this, the modulus of the reflection coefficient from the earth is taken equal to 1, and its phase as \(180^\circ\).

Fig. 33. Path of rays undergoing a single reflection from a flat layer

Considering all the rays parallel, we find that

\[ \begin{aligned} r_1 &= \rho_1 - (h+z)\sin\theta_1; \qquad r_2 = \rho_1 + (h+z)\sin\theta_1;\\ r_3 &= \rho_1 - (h-z)\sin\theta_1; \qquad r_4 = \rho_1 + (h-z)\sin\theta_1, \end{aligned} \]

INFLUENCE OF THE TROPOSPHERE ON THE STABILITY OF RADIO-WAVE RECEPTION

where

\[ \rho_1=\sqrt{R^2+(2H)^2}\simeq R \quad \text{and} \quad \sin\theta_1=\frac{2H}{\rho_1}. \]

This gives:

\[ E_1=\frac{300\sqrt{P_{\Sigma}}}{R}\cdot F_1 \cdot 2\sin(kh\sin\theta)\cdot 2\sin(kz\cdot\sin\theta)\cdot e^{-j(\omega t-k[[unclear: exponent term]])}, \]

whence approximately we have:

\[ E_{10}\simeq \frac{300\sqrt{P_{\Sigma}}}{R}F_1\cdot 2\sin\left(\frac{2\pi hH}{\lambda R_1}\right)\cdot 2\sin\left(\frac{2\pi zH}{\lambda R_1}\right), \]

which corresponds to formula (14) for a spherical earth.

Fig. 34. Case of double reflection

Similarly, for the field at point \(B\), produced by a ray reflected \(n\) times from the layer, one can obtain:

\[ E_{n0}=\frac{300\sqrt{P_{\Sigma}}}{\rho_n}(-1)^{n+1}F_n^n\cdot 2\sin(kh\sin\theta_n)\cdot 2\sin(kz\sin\theta_n). \]

Here (see, for example, the construction for \(n=2\) in Fig. 34)

\[ \rho_n=\sqrt{R^2+(n2H)^2}\simeq R \quad \text{and} \quad \sin\theta_n=\frac{n2H}{\rho_n}. \]

The total field produced at point \(B\) by the whole set of such rays is approximately determined as

\[ E_0=\frac{300\sqrt{P_{\Sigma}}}{R}\,4\sum_{n=1}^{n=\infty}(-1)^{n+1}F_n^n\sin(n\alpha)\cdot \sin(n\beta). \]

where

\[ \alpha=\frac{4\pi hH}{\lambda R_1} \quad \text{and} \quad \beta=\frac{4\pi zH}{\lambda R_1}. \]

For simplicity, let us put \(F_n=F_1\) and \(h=z\); then

\[ E_0=\frac{300\sqrt{P_{\Sigma}}}{R} \sum_{n=1}^{n=\infty}(-F)^n \left(2-e^{j2\alpha n}-e^{-j2\alpha n}\right)= \]

\[ =\frac{300\sqrt{P_{\Sigma}}}{R}\cdot F_1\cdot 2 \left(\frac{1-F_1}{1+F_1}\right) \left(\frac{1-\cos 2\alpha}{1+F_1^2+2F\cos 2\alpha}\right). \]

Since the reflection coefficient \(F_1\) is very small, in the limit

\[ E_0=\frac{300\sqrt{P_\Sigma}}{R}F_1\cdot 4\sin^2\left(\frac{4\pi h h'}{\lambda R}\right)=E_1. \]

Thus, in an approximate calculation of the field at the point \(B\), in view of the smallness of the reflection coefficients from the layer (an effect which is further intensified by the decrease of \(F\) as the angle of incidence increases), one may indeed confine oneself to taking into account only rays of the “first” order. This result may also be extended to the case of a spherical earth, on which we shall not dwell. We also note that the question of the propagation of radio waves between two parallel planes (the earth and an ionized layer) was considered by Kenrick \(^{77}\).

Appendix II

The field at a receiving point located within the horizon may be approximately represented (see \(^{31}\), pp. 46, 60 and 42) as

\[ E=\frac{300\sqrt{P_\Sigma}}{R}\,2\cdot \sin\left(\frac{2\pi\Delta r}{\lambda}\right), \]

where \(\Delta r=\dfrac{2h'z'}{R}\). Hence, proceeding from the condition

\[ \frac{dE}{d(\Delta r)}=0, \]

we find that the field \(E\) is least “sensitive” to variations of the path difference \((\Delta r)\) when \(\Delta r=\dfrac{\lambda}{2}\).

Since the initial formula for \(E\) was obtained under the assumption of a phase jump upon reflection from the earth of \(180^\circ\), at the indicated value \(\Delta r=\dfrac{\lambda}{2}\) both rays at the receiving point are in phase. This corresponds to the maximum field at the receiving point; in this case the “axis of the lobe” of the radiation pattern (see \(^{31}\), p. 39) passes through the receiving point.

We also indicate that, for equal heights \(h\) and \(z\), their values corresponding to a given path difference (with allowance for the mean refraction according to the concept of the equivalent radius) are determined as

\[ h=z=\frac{R^2}{8a_e}+\sqrt{\frac{R}{2}\Delta r}. \]

The table gives the values of \(h\) for several \(R\) and \(\lambda\), for the path difference \(\Delta r=\dfrac{\lambda}{2}\), if one assumes that \(a_e=ma=\dfrac{4}{3}a\).

\(\lambda\), m	0.3	0.6	1	3	6
\(R=20\) km	\(R=20\) km	\(R=20\) km	\(R=20\) km	\(R=20\) km	\(R=20\) km
\(h\), m	44.6	60.7	76.6	128	177.9
\(R=40\) km	\(R=40\) km	\(R=40\) km	\(R=40\) km	\(R=40\) km	\(R=40\) km
\(h\), m	78.3	100.9	123.5	196.5	267.5

If one takes \(m\) somewhat different from \(4/3\), for example \(5/4\), the result will change only very little.

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1. See [^31], Table 18 (p. 257), concerning the experiments of Joua[^54] (France), Marconi[^55] (Italy), VEI[^32] (USSR), Hershenberger[^56] (USA), Lindenblad[^57] (USA), and NK Svyazi of the USSR, as well as some data on the experiments of Scholz[^58] (Germany), Colster[^59] (USA), and Trevor and George[^60], cited there on pp. 268 and 269. ↩↩↩↩↩↩↩↩

2. For more details on this work see [^31], p. 268. ↩↩↩↩

3. See Uspekhi fizicheskikh nauk, 23, 405, 1940. ↩↩

4. Reference number as printed in the source. ↩