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Volume 77, 1948-49
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A Preliminary Study Of Some Results From The Radio-Meteorological Investigation Conducted In Canterbury.

Introduction: The propagation of ultra high frequency electro-magnetic radiation near the surface of the earth is dependent to a large degree on the distributions of temperature and humidity in the first few hundred feet of the atmosphere. Some of the theoretical aspects behind this propagation are briefly considered, and it is shown how a particular climatic modification of the lower atmosphere can radically influence the propagation, more particularly in the region beyond the geometrical horizon of the transmitter, a space normally associated with the relatively weak diffraction field.

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The particular modification is that which occurs when warm dry air flows out from land over a colder sea. The resulting distributions of refractive index with height in the first few hundred feet have been measured, and are such as to cause abnormally strong signals to be received well beyond the horizon of a transmitter.

Orthodox Propagation: In an atmosphere in which the air is well mixed, the distributions of temperature and humidity with height result in a linear decrease of refractive index with height for all relevant values of height, of such a magnitude that the downward curvature of a ray in this space is approximately one quarter of the earth's curvature, a factor which increases the effective horizon of a source above the earth's surface. Utilising the concept of modified refractive index which permits the treatment of the earth's surface as a flat one, it follows that in a well-mixed atmosphere the modified refractive index increases linearly with height.

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Fig. 1—The Dependence of Refractive Index μ and Modified Refractive Index M in well-mixed Atmosphere.

At centimetre wave-lengths, both the earth and sea behave as almost perfect reflectors, and, therefore, under normal conditions the field distribution some distance from a transmitter will depend on height in the manner shown in Fig. 2.

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Fig. 2—The Dependence of Field Strength E on height.

Above the horizon, the normal lobe structure is evident, due to the interference of the direct and reflected rays. The calculation of the lobe positions must take into account the slight bending of rays in the atmosphere. The fact that the minima of field strength are not zeros is due to a divergence factor introduced by the sphericality of the earth.

Below the horizon is the diffraction field, decreasing in intensity as the height becomes smaller. The scale of the lobe structure and the diffraction curve will, of course, depend on the height of the transmitter, the wave-length of the radiation, and the separation distance of transmitter and receiver.

It will be seen below how it is possible to achieve well below the horizon field strengths comparable with that existing above the horizon.

Anomalous Propagation: It is well known that certain temperature and humidity distributions in the lower atmosphere result in a modified refractive index distribution which, over a limited height called the height of the radio duct, is conducive to the downward bending of rays. The dependence of modified refractive index on these two meteorological quantities is such that a temperature excess and a specific humidity deficit near the surface will provide the necessary variation in modified refractive index.

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Fig. 3—The Potential Temperature and Specific Humidity Distributions Resulting in the Formation of a Surface Duct.

Employing the ray theory of propagation, it can be shown* that under these conditions a certain amount of the energy from the transmitter is trapped in the duct, and will result in abnormally strong fields well beyond the horizon, if, of course, the duct extends as far as this.

Mode Theory of Propagation in a Duct: The ray-theory treatment of super-refraction in the atmosphere, which consists in considering rays being refracted down to the surface, reflected, refracted back once more to the surface, reflected, and so on, immediately recalls the propagation of ultra high frequency radiation along a waveguide.

There exists between the two situations quite a useful analogy. The passage of a transverse electric wave of the Hon type down a waveguide may be regarded as being accomplished by the successive reflection of a plane wave at two opposite walls of a guide. The pattern of electric field or distribution of field across the guide is determined by the guide dimensions, the wave-length, and the angle of reflection. The boundary conditions permit only a finite number of discrete angles of reflection to be assumed; to each there corresponds a certain field pattern, called a mode of propagation. Some of these distributions are shown in Fig. 4.

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Fig. 4—Modes of Propagation in Waveguide.

Now consider the propagation of horizontally polarised radiation in an atmosphere, the modified refractive index of which decreases linearly with height. A plane wave starting at some small angle will be refracted and ultimately return to the earth's surface, where it is reflected. Compare this process with the waveguide mechanism. The surface of the earth or sea acts like the bottom side of the waveguide, and the function of the other side is performed by the refractive properties of the medium.

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Fig. 6—Modes of Propagation in an infinite Radio Duct.

[Footnote] * See succeeding paper.

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Exactly as in the waveguide there exist definite modes of propagation or patterns of electric field which travel immediately above the surface; some of the distributions are shown in Fig. 5.

At the earth's surface, the field is zero for all the modes, but there is no well-defined upper limit to the field as there is in a waveguide. Another essential difference between the two cases is that the energy associated with each mode is not principally confined to the same track-width for all modes. The track-width increases with mode number, but depends also on the wave-length and the la pserate of refractive index.

The atmosphere chosen as an example is a very artificial one, since the region of negative dM/dh is usually confined to a few hundred feet. Because of the dependence of track-width on wave-length and mode number, only the low order modes for the smaller wave-lengths will be propagated without attenuation. Higher modes at these wave-lengths and all modes of greater wave-lengths will not be “trapped.” the energy associated with them leaking away from the top of the duct, the leakage increasing with mode number and wave-length. This explains the fairly frequent occurrence of anomalous propagation at centimetre wave-lengths.

Mathematical Treatment: The above solution is not directly applicable to the problem in general because, even if we assume stratification, i.e., the same M profile existing for all distances, the dependence of modified refractive index on height is not linear or bilinear. However, an attack via mode theory is still extremely powerful, and although entailing not inconsiderable numerical work, the first and second mode solutions have been obtained for some simple profiles.

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For horizontally polarised radiation, the electric field vector E is contained in the wave equation ò2E/òx2 + ò2E/òh2 + k2μ2 E = O (1)

Where × is the horizontal co-ordinate, h is the height, k is the wave number, and μ is the modified refractive index, dependent in some way upon the height h.

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A solution of this equation may be obtained in the form E = ΣanUn (h) exp (−i k μ0 × cos θn) (2)

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where μ0 is the value of μ when h = o. Each term of this solution in series represents a mode of propagation, and Un (h) is a solution of the one dimensional wave equation d2Un/dh2 + k22—μo2cos2θn) Un = O (3)

The boundary conditions Un (O) = O and Un(h)→ O as h → ∞ permit only discrete values of cos θn. These, the eigenvalues of the problem, may be real or complex, corresponding to trapped or leaky modes. Their computation and the evaluation of the associated eigenfunctions Un (h) provide a difficult problem for all distributions of refractive index except the very simplest.

Undoubtedly, the assumption of stratification provides a convenient first approximation, but it appears from the few observations we have obtained on advection ducts that the distribution of refractive index with height varies not inconsiderably with offshore distance, and that a final solution will have to take into account this dependence of modified refractive index on the two variables × and h.

A Typical Advection Duct: That the types of temperature, humidity, and refractive index distributions which have been discussed do exist under certain meteorological conditions, and that they result in pronounced superrefraction at centimetre wave-lengths, is evident from a set of observations obtained in Canterbury in February of this year.

A typically warm, dry, north-west wind was blowing steadily over the area of observations for the duration of the measurements.

The temperature modification introduced by the sea surface may be studied in Fig. 6, which is a cross-section perpendicular to the coast and approximately in the direction of the wind.

The average potential temperature of the air-mass over the land is 75° F., whereas over sea the potential temperature varies from 59° F. immediately above the surface to 75° F. for the undisturbed upper layers. This modification, sharp at first, occupies a greater height as the offshore distance increases, but is com-

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Fig. 6—Isopleths of Potential Temperature—°F.

plicated by a variation in sea temperature. For the first 40 kms., the sea temperature is 59° F., then it drops steadily to a value of 54° F. at 80 kms., beyond which it appears to be maintained.

The effect of evaporation from the surface into the lower layers of the air-mass is shown in Fig. 7, similar to the previous diagram, but with specific humidity replacing potential temperature. The modification in this case, however, is not so well defined.

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Fig. 7—Isopleths of Specific Humidity—Q gms./kgm.

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Fig. 8.—Isopleths of Modified Refractive Index—M.

The resulting distribution of refractive index is given in Fig. 8. Above 700 ft., the modification caused by the sea surface is negligible even at great ranges, i.e., 120 mk. The duct height, i.e., the height of the M inversion, increases rapidly it offshore distance, reaching a maximum height of 300 ft. at 60 mk., which is maintained for another 60 mk. at least.

The coast is a point of discontinuity, since the surface temperature and humidity both change radically here. With the variation of duct height with distance, there is also a considerable change in the shape of the M curve. Just

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Fig. 9.—Variation of Duct Shape with Distance Offshore.

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offshore, the duct is very intense, but also very low; by intensity, we choose to mean the difference between the surface value of M and the minimum value. As the distance increases, the intensity lessens. This is seen in Fig. 9.

From calculations made for M profiles of roughly similar dimensions, it is reasonable to suppose that such a radio duct as the one encountered above is capable of considerable trapping of centimetre radiation. Such a prediction is completely borne out by field strength measurements made, this day well beyond the geometrical horizon. These are given in Fig. 10, where for each of the four transmitters the field strength in decibels above noise is plotted against height. In order to have a useful absolute measure of these field strengths, the free space field strength was estimated by observing the lobe structure obtained from the aircraft flying at constant height away from the transmitter. The level of free space field in decibels above noise is inserted in each of the height gain curves of Fig. 10. Unfortunately, there is inherent in our method of observation a small range variation in the height gain curves, but this we believe to be outweighed by other considerations.

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Fig. 10.—Height Gain Curves obtained on Slant Descent. Range Change—2,000–1,000–50 ft.

Referring to the first height gain curve in Fig. 10, which is for a transmitter 29 ft. above sea-level operating at a wave-length of 9.3 cm., it is evident that at a distance of 80 km. the field above the horizon corresponds roughly to a lobe pattern, but below the horizon, instead of decreasing rapidly, a fairly constant signal level is maintained down to 100 ft., where there exists a small maximum.

The same effect is achieved on the other transmitters, one at the same height operating at 3.2 cm., and a similar pair of transmitters at a height of 88.5 ft. above sea-level. The absence of any radical change in the height gain curves due to the change of transmitter height is not surprising, since both transmitters are well within the radio duct. Only if the transmitters were placed well above the duct would the field strength well beyond the horizon be diminished appreciably.

Although it is not possible at this stage to predict completely the field strength variation with height associated with a given field of modified refractive index, we are permitted one or two general deductions.

Were propagation beyond the horizon performed by the first mode alone, we would expect to have a maximum somewhere in the duct, e.g., 100 ft., but above the duct the signal would be expected to fall off rapidly with height, only increasing again as the horizon is approached. This decrease above the duct is

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non-existent in our case. This may be due to the presence of a number of trapped modes, or the fact that the energy associated with the large number of leaky modes will escape from the duct into the region above it. Of course, the decrease in intensity of the duct with distance will undoubtedly result in some departure from the classical picture of mode propagation in a radio duct.

Some General Conclusions: It appears that under north-west conditions in Canterbury, the necessary changes of temperature and humidity near the sea surface are sufficient to cause very considerable superrefraction on wave-lengths of 3 and 10 cm.

The excess of upper air potential temperature over the surface temperature is frequently of the order of 15–25°F. and the humidity deficit often 5 gms/kgm.

Although it has been our misfortune to examine only a few advective situations, it would appear that the radio duct extends to at least 150 km. offshore, although its effect, i.e., its ability to cause trapping of radio energy, is much weakened at this stage. The maximum height of the modified refractive index inversion is of the order of 300–500 ft., and is attained within the first 40–60 km.

Obviously such figures will vary with the wind speed and wind profiles, the warmth and the dryness of the nor'-wester, and the sea temperature, but they are, we think, generally illustrative of the conditions which do occur. The situation is sometimes complicated by the presence of an onshore sea breeze, but this is usually confined by the offshore wind to a few miles each side of the coastline, and the effect on propagation is small. A more complex set of conditions is provided when the north-west wind is kept aloft by a north-east wind which may extend in height up to a thousand feet.

It is hoped that it will be possible to obtain an accurate correlation between the modified refractive index distribution and the field strength dependence on height. This should make possible the prediction of propagation curves for different frequencies and for various advective situations, and a correct assessment of the modifications introduced by some of the simpler complicating factors mentioned above.

Acknowledgment: For the initial portion of this article, the author is indebted to H. G. Booker and G. G. Macfarlane, of the Telecommunications Research Establishment, Naiver, England.