where dM / dh is the vertical gradient of the modified refractive index and Ah is the thickness of the duct, corresponding to the height above the emission antenna of the upper boundary layer of the duct.

The maximal trapping angle increases rapidly as the refractive index gradients falls below - 157 N units/km or when the thickness of the duct increases. Recommendation ITU-R P.834 presents curves giving the maximum trapping angle depending on the refractive index and on the thickness of the duct (ITU-R 1995).

It can therefore be seen that the presence of a propagation duct influences the level of the received signal: although the multiple paths cause a fading of the signal, however, as far as interferences are concerned, the most important property of this mode of propagation is the possible enhancement of the signal.

The previously described case of duct propagation could be reduced to the consideration of two main types: surface ducts and elevated ducts. However, the complexity of atmosphere results in the possibility for multiple layers to exist. Although the refractive index varies on average with the atmospheric pressure in altitude, local variations of the refractive index, caused by irregularities in the temperature and in the relative humidity of the air, may also appear. These irregularities are themselves due to the temperature inversion occurring at sunrise and to anticyclonic subsidence. The thin reflective layers thus created are also called sheets. The reflective capacity of these layers depends on the gradient of their refractive index. These layers may extend over several tens of square kilometres, while their thickness may reach an order of 100 metres. As an example, the altitude at which these layers form in the United Kingdom is approximately 1.4 km (Lane 1965).

The probability for such layers to exist is higher than the probability of a tropo-spheric radio duct. This results from the fact that while a strong variation of the refractive index may be at the origin of these layers, an additional condition for the formation of a tropospheric duct is that the thickness of the layers must be sufficient for the waves to be guided.

From a macroscopic point of view, the atmosphere exhibits an horizontal stratification: accordingly, tropospheric layers can be regarded as being roughly plane surfaces. These surfaces, however, can be deformed through a number of different mechanisms, for instance the vertical gradients of the wind, turbulences or the movement of vertical layers. The irregularities thus superimposed to the plane surface are of two types:

- primary irregularities, with a size of the order of the metre, which are generated by microscopic flows in the vicinity of the surface,

- secondary irregularities, with a size of the order of ten metres, i.e. an order of magnitude larger than the primary irregularities.

Over very broad layers, sinusoidal irregularities with wavelengths of a few kilometres can be sometimes observed (Lane 1965).

The reflection of waves at layers of this type may occur in two different ways:

- either by specular reflection if the separation surface between the layers does not present any irregularity,

- either by scattering reflection in the more frequent case where the interface is irregular.

Small variations in the characteristics of these layers, either in the quality of their surfaces or in their position, generate fluctuations of the level of the received signal. The signature of this propagation mechanism reveals the existence of both slow and fast variations, to which may be added scintillations due to tropospheric scatter.

The combination of reflection and tropospheric refraction may result in relatively high signal levels beyond the radio horizon. Refraction induces effects on the curvature of the ray paths, which result in the reduction of the incidence angle of the rays with the reflective layer and enhance the conditions of reflection. As regards the problem of interferences, it can therefore be seen that a transmitter may reach points located beyond the geometrical horizon.

Several different theories have been developed in order to account for the phenomena of reflection at these layers. The simplest such theories are very similar to Fresnel theory with the difference however that they take into account the irregularities that these layers present. For this purpose a reduction factor p is introduced through the following definition (Boithias 1987):

where Ah is the standard deviation of the height distribution of the irregularities inside the layer, which is assumed to be a Gaussian distribution, while 9 is the incidence angle of the ray with the average surface.

An effect of the combination of reflection and refraction is that when the angle of incidence decreases, the p reduction factor tends towards the unit and accordingly the higher the reflection factor becomes. Even though the theoretically predicted value of the reflection factor is no higher than 1, this mechanism of propagation still represents a real threat in the case of relatively important percentages of time.

5.6.5 Diffraction

The presence of an obstacle, and more specifically of the Earth, between a transmitter and a receiver may result in a received signal with a relatively high power. This is due to the fact that the direct signal and the signal emitted back by the obstacle may both be received at the reception point. This phenomenon, which cannot be explained by geometrical optics, is known as diffraction.

Although the study of diffraction phenomena is primarily concerned with boundary condition problems, the surface complexity of actual obstacles makes this study extremely difficult.

In the case of waves propagating at a close distance from the Earth, the main obstacle is the Earth itself. The higher the frequency is, the higher is the influence exerted by the irregularities of the ground. In diffraction studies, two well-studied models for the diffraction by obstacles exist: single knife-edge diffraction and rounded obstacle diffraction. These two models can then be combined for describing more complex cases. There exist other models of diffraction intended at describing more complex forms of diffraction, in particular the models developed by Deygout, Millington, Epstein and Peterson (Boithias 1987). More detailed information on this subject can be found in Appendix K devoted to the methods used for determining diffraction.

The term of spherical diffraction propagation is applied to the propagation of waves along paths without any significant diffraction edge. In this case, the amplitudes of the received signals are in general much lower than the amplitudes of the received signals in the case of propagation by knife-edge diffraction. However, due to the stability of signals propagated by spherical diffraction along favourable paths, these signals may still have a significant amplitude in the absence of ducting, i.e. for instance for percentages of time higher than 20 percent.

A link where the transmitter and the receiver are in direct visibility is not necessarily a line-of-sight link: this essentially depends on the value of the refractive index. More specifically, and as represented in Fig. 5.19, when the conditions for subrefraction are fulfilled, waves may graze the ground and be diffracted by possible obstacles present along the path.

In order to study diffraction phenomena, Fresnel introduced a division of space into different regions which have since been referred to under the name of Fresnel ellipsoids. This subject is addressed at more depth in Chap. 3. Let us here simply recall that if the first Fresnel ellipsoid, or at least 60 percent of its surface, is clear, then it is not necessary to take into account the effects induced by diffraction.

As in the case of tropospheric reflection, the combination of diffraction and refraction is a possible cause of interferences which is not to be neglected. Fig. 5.20 presents an example of a realistic path which clearly shows that the attenuation due to diffraction is inversely proportional to the gradient of the refractive index.

Fig. 5.19. Refraction and path profile: occultation of the radio wave in subrefraction

Refracthrity gradient dN/dh (N-units/km)

Fig. 5.20. The attenuation due to spherical diffraction at the 4 GHz frequency as a function of the gradient of the refractive index (CCIR 1990)

Refracthrity gradient dN/dh (N-units/km)

Fig. 5.20. The attenuation due to spherical diffraction at the 4 GHz frequency as a function of the gradient of the refractive index (CCIR 1990)

Under the assumption of a spherically stratified atmosphere with a vertical gradient of the refractive index, the effects induced by refraction can be represented by using the effective Earth radius, since the latter is indeed dependent on the gradient of the refractive index. The trajectories followed by waves can thus be represented by straight lines.

The effective Earth radius is given by the equation:

where k is the effective Earth radius factor, while a is the effective Earth radius.

For a standard atmosphere, AN /Ah = - 39 N units/km, while k = 4/3 and Re = 8500 kilometres.

Fig. 5.21 shows the representation of the radio horizon obtained by using the effective Earth radius. If ht is the height of the emitting antenna, then the distance to the radio horizon is given by the following equation:

Hence, the larger the k factor is, the larger is the distance to the radio horizon. Signals obtained in the conditions of spherical diffraction increase and decrease without undergoing any significant fluctuations. As the level of the signals is not as high as in the case of ducting propagation, a phenomenon of tropospheric scatter may still often appears, which means that a superposition of the effects induced by these two propagation mechanisms occurs.

The probability that such a phenomenon occurs is higher than the probability of a duct, especially in the case of short terrestrial links.

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