The parameters of a measurement system (meteorology).
Influence of atmospheric refraction on the propagation of electromagnetic waves has been studied from the beginnings of radio wave technology (Kerr, 1987). It has been proved that the path bending of electromagnetic waves due to inhomogeneous spatial distribution of the refractive index of air causes adverse effects such as multipath fading and interference, attenuation due to diffraction on the terrain obstacles or so called radio holes (Lavergnat & Sylvain, 2000). These effects significantly impair radio communication, navigation and radar systems. Atmospheric refractivity is dependent on physical parameters of air such as pressure, temperature and water content. It varies in space and time due the physical processes in atmosphere that are often difficult to describe in a deterministic way and have to be, to some extent, considered as random with its probabilistic characteristics.
Current research of refractivity effects utilizes both the experimental results obtained from in situ measurements of atmospheric refractivity and the computational methods to simulate the refractivity related propagation effects. The two following areas are mainly addressed. First, a more complete statistical description of refractivity distribution is sought using the finer space and time scales in order to get data not only for typical current applications such as radio path planning, but also to describe adverse propagation in detail. For example, multipath propagation can be caused by atmospheric layers of width of several meters. During severe multipath propagation conditions, received signal changes on time scales of minutes or seconds. Therefore, for example, the vertical profiles of meteorological parameters measured every 6 hours by radiosondes are not sufficient for all modelling purposes. The second main topic of an ongoing research is a development and application of inverse propagation methods that are intended to obtain refractivity fields from electromagnetic measurements.
In the chapter, recent experimental and modelling results are presented that are related to atmospheric refractivity effects on the propagation of microwaves in the lowest troposphere. The chapter is organized as follows. Basic facts about atmospheric refractivity are introduced in the Section 2. The current experimental measurement of the vertical distribution of refractivity is described in the Section 3. Long term statistics of atmospheric refractivity parameters are presented in the Section 4. Finally, the methods of propagation modelling of EM waves in the lowest troposphere with inhomogeneous refractivity are discussed in the Section 5.
2. Atmospheric refractivityAdvertisement
2.1. Physical parameters of air and refractivity formula
The refractive index of air
It can be simply demonstrated, based on the Debye theory of polar molecules, that refractivity can be calculated from pressure
where for the saturation vapour above liquid water
It is seen in Fig.1a where the dependence of the refractivity on temperature and relative humidity is depicted that refractivity generally increases with humidity. Its dependence on temperature is not generally monotonic however. For humidity values larger than about 40%, refractivity also increases with temperature.
During standard atmospheric conditions, the temperature and pressure are decreasing with the height above the ground with lapse rates of about 6º C/km and 125 hPa/km (near ground gradients). Assuming that relative humidity is approximately constant with height, a standard value of the lapse rate of refractivity with a height
2.2. EM wave propagation basics
Ray approximation of EM wave propagation is convenient to see the basic propagation characteristics in real atmosphere. The ray equation can be written in a vector form as:
where a position vector r is associated with each point along a ray and
Three typical propagation conditions are observed depending on the numerical value of the gradient. If d
Sub-refractive atmospheric conditions occur when the refractivity gradient has a significantly larger value, super-refractive conditions occur when the refractivity gradient is well below the standard value of -40 N-units/km. During sub-refractive atmospheric conditions, the effective Earth radius
In principle, the EM wave propagation characteristics during clear-air conditions are straightforwardly determined by the state of atmospheric refractivity. Nevertheless, atmospheric refractivity varies in time and space more or less randomly and full details of it are out of reach in practice. Therefore the
3. Measurement of refractivity and propagationAdvertisement
3.1. Measurement setup
A propagation experiment focussed on the atmospheric refractivity related effects has been carried out in the Czech Republic since November 2007. First, the combined experiment consists of the measurement of a received power level fluctuations on the microwave terrestrial path operating in the 10.7 GHz band with 5 receiving antennas located in different heights above the ground. Second, atmospheric refractivity is determined in the several heights (19 heights from May, 2010) at the receiver site from pressure, temperature and relative humidity that are simultaneously measured by a meteo-sensors located on the 150 meters tall mast. Refractivity is calculated using (2) – (4). Figure 2a shows the terrain profile of the microwave path.
The distance between the transmitter and receivers is 49.8 km. It can be seen in Fig. 2a a terrain obstacle located about 33 km from the transmitter site. The height of the obstacle is such that about 0% of the first Fresnel ellipsoid radius of the lowest path (between the transmitter antenna and the lowest receiver antenna) is free. It follows that under standard atmospheric conditions (
|Heights of meteorological sensors||5.1 m, 27.6 m, 50.3 m, 75.9 m, 98.3 m, 123.9 m, 19 sensors approx. every 7 m (from May 2010)|
|Pressure sensor height||1.4 m|
|Temperature/humidity sensor||Vaisala HMP45D, accuracy ±0.2°C, ±2% rel. hum.|
|Pressure sensor||Vaisala PTB100A, accuracy ±0.2 hPa|
|TX tower ground altitude||258.4 m above sea level|
|TX antenna height||126.3 m|
|TX output power||20.0 dBm|
|Path length||49.82 km|
|Parabolic antennas||diameter 0.65 m, gain 33.6 dBi|
|RX dynamical range||"/ 40 dB|
|RX tower ground altitude||188.0 m above sea level|
|RX antennas heights||51.5 m, 61.1 m, 90.0 m, 119.9 m, 145.5 m|
|Est. uncertainty of received level||±1 dB|
3.2. Examples of refractivity effects
In order to get a better insight into atmospheric refractivity impairments occurring in real atmosphere, several examples of measured vertical profiles of temperature, relative humidity, modified refractivity and of received signal levels are given. The modified refractivity
Figure 3 shows the example of radio-meteorological data obtained during a very calm day in autumn 2010. The relative received signal levels measured at 51.5 m (floor 0), 90.0 m (floor 2) and at 145.5 m (floor 4) are depicted. The lowest path (floor 0) is attenuated of about 6 dB due to diffraction on a path obstacle. The situation is atypical since the received signal level is very steady and does not fluctuate practically. The vertical gradient of modified refractivity has approximately the same value (≈ 110 M-units/km or -47 N-units/km) during the whole day, the propagation conditions correspond to standard atmosphere.A more typical example of measured data is shown in Fig. 4. Temperature and relative humidity change appreciably with height and in time. Specifically, temperature inversion is seen before 4:00 and after 20:00, the standard gradient takes place in the middle of the day. The received signal level recorded on the lowest path shows a typical enhancement at the beginning and at the end of the day which is caused by super-refractive propagation conditions. On the other hand the signal received at the higher antennas fluctuates mildly around 0 dB with more pronounced variations of the signal in the morning and at night.
Sub-refractive propagation conditions were observed between 2:00 and 4:00 on 14 October 2010 as shown in Fig. 5. One can see that increased attenuation due to diffraction on the path obstacle appears on the lowest path (floor 0) at that time. This well corresponds with the sub-refractive gradient of modified refractivity observed; see the lower value of d
4. Refractivity statistics
As already mentioned, the physical processes in troposphere are complex enough to allow only statistical description of spatial and temporal characteristics of atmospheric refractivity. Nevertheless the statistics of important refractivity parameters such as an average vertical gradient are extremely useful in practical design of terrestrial radio paths when the long term statistics of the received signal have to be estimated, see (Rec. ITU-R P.530-12, 2009).
4.1. Average vertical gradient of refractivity
The prevailing vertical gradient of refractivity can be regarded as the single most important characteristics of atmospheric refractivity. According to (6), it is related to the effective Earth radius discussed above and it specifically determines the influence of terrain obstacles on terrestrial radio propagation paths. The examples of measured vertical profiles presented in the previous section show that the near-ground refractivity profile evolution is complex enough to not be described by only a single value of the gradient. The question arises what should be considered as a prevailing vertical gradient at a particular time. The gradient value is usually obtained from the refractivity difference at fixed heights, e.g. at 0 and 65 meters above the ground (Rec. ITU-R P.453-9, 2009). If more accurate data is available, the prevailing vertical gradient of refractivity can be calculated using a linear regression approach.
Two year data (2008-2009) of measured vertical profiles were analysed by means of linear regression of refractivity in the heights (0 – 120 m) and the statistics of the vertical gradient so obtained were calculated. The results are in Fig. 7a where the annual cumulative distribution functions of the gradient are depicted. The quantiles provided by ITU-R datasets are also shown for comparison. It is clear that extreme gradients are less probable in reality than predicted by ITU-R. Linear regression tends to filter out the extreme gradients (otherwise obtained from two-point measurements) which do not fully represent the vertical distribution as a whole.
Taking into account the importance of the gradient statistics for the design of terrestrial radio path, it seems desirable to have a suitable model. Several models of the gradient statistics were proposed, see (Brussaard, 1996), that can be fitted to measured data. Since they are often discontinuous in the probability density, they can be thought to be little unnatural. One can see in Fig. 7b where the two-year cumulative distribution is shown that the distribution consists of three parts: the part around the standard (median) gradient and two other parts – tails. Therefore the following model of the probability density
4.2. Ducting layers
Although the ducting layers appearing in the first several tens or hundreds meters above the ground have significant impact on the propagation of EM waves on nearly horizontal paths, surprisingly little is known about their occurrence probabilities or about their spatial/temporal properties (Ikegami et al., 1966). This is true especially in the lowest troposphere where the usual radio-sounding data suffers from insufficient spatial and also time resolution. In the following, the parameters of ducting layers observed during the experiment are analysed by means of the modified Webster duct model.An analytic approach to the modelling of refractivity profiles was proposed in (Webster, 1982). The refractivity profile with the height
where the refractivity
The above model was fitted to the refractivity profiles measured in between May and November 2010. More than 3 105 profiles were analysed and related model parameters were obtained. Figure 9 shows two examples of 1-hour measured data and fitted models. Significant dynamics is clearly seen in the evolving elevated ducting layers. It is also clear from the examples in Fig. 9 that the model is not able to capture all the fine details of measured profiles but it serves very well to describe the most important features relevant for radio propagation studies. Sometimes, the part or the whole ducting layer is located above the measurement range and so it is out of reach of modelling despite its effect on the propagation might be serious. This should be kept in mind while studying the statistical results presented below.
Figure 10 shows the empirical cumulative distributions of duct model parameters obtained from the fitting procedure. The medians (50% of time) of duct parameters can be read as
Important interrelations between duct parameters are revealed by empirical joint probability density functions (PDF) presented in Fig. 11 – 15. The 2D maps show the logarithm of joint PDFs of all combinations of 5 parameters of the duct model (10). In these plots, dark areas mean the high probability values and light areas mean the low probability values. It is generally observed that there are certain preferred areas in the parameter space where the combinations of duct parameters usually fall in. For example, it is seen in Fig. 13a that the absolute value of the negative duct depth is likely to increase with the increasing gradient
5. Modelling of EM waves in the troposphere
Several numerical methods have been used in order to assess the effects of atmospheric refractivity on the propagation of electromagnetic waves in the troposphere. They can be roughly divided into two categories - ray tracing methods based on geometrical optics and full-wave methods. The ray tracing methods numerically solve the ray equation (5) in order to get the ray trajectories of the electromagnetic wave within inhomogeneous refractivity medium. The ray tracing provides a useful qualitative insight into refraction phenomena such as bending of electromagnetic waves. Its utilization for quantitative modelling is limited to conditions where the electromagnetic waves of sufficiently large frequency may be approximated by rays. Geometrical optics description is known to fail at focal points and caustics where the full-wave methods provide more accurate results.
The full-wave numerical methods solve the wave equation that is a partial differential equation. Among time domain techniques, finite difference time domain (FDTD) based approaches were proposed (Akleman & Sevgi, 2000) that implement sliding rectangular window where 2D FDTD algorithm is applied. Nevertheless, tropospheric propagation simulation in frequency domain is more often. In particular, there is a computationally efficient approach based on the paraxial approximation of Helmholtz wave equation, so called Parabolic Equation Method (PEM), which is the most often used full-wave method in tropospheric propagation.
5.1. Split step parabolic equation method
We start the brief summary of PEM (Levy, 2000) with the scalar wave equation for an electric or magnetic field component
is the modified refractive index which takes account of the Earth’s radius
in equation (12) to obtain:
Paraxial approximation is made now. The field
and the 1/(2
An elliptic wave equation is therefore simplified to a parabolic equation where near horizontal propagation is assumed. This equation can be solved by the efficient iterative methods such as the Fourier split-step method. Let us assume the modified refractivity
where Fourier transform is defined as:
From (18), we obtain:
and we get the formula for step-by-step solution:
The field in the next layer
Fourier transformation is applied in z-direction and the variable
The assumption that
5.2. Application example and comparison with measured data
The parabolic equation method outlined above has been applied frequently to investigate the propagation characteristics on terrestrial (and also on Earth - space) paths under the influence of different refractivity conditions (Barrios, 1992, 1994; Levy, 2000) including the ducting layers described in the section 4.2. Users agree the method gives reliable results provided all the relevant details of terrain profile and of refractivity distribution are known and modelled correctly. This is however not always the case in practice. It is believed that the modelling results have to be compared with real world data whenever possible in order to validate the method under different propagation conditions and to know more about the expected errors due to incomplete knowledge of propagation medium.
Let us illustrate the particular example of conditions where the parabolic equation method performs successfully regardless the fact that refractivity profile along the propagation path is only roughly estimated. Figures 16a and 16b show the results of PEM propagation simulation performed using refractivity gradients measured during the 4th of November, 2008 at the receiver site. Sub-refractive conditions that occurred early morning caused a significant diffraction fading of more than 20 dB on the two lowest paths see Fig. 16b. On the other hand, the higher paths (receiving antennas located at 90 m and above) were not affected by diffraction effects.
The results shown in Fig. 16b confirm that a very good agreement between PEM simulation and measurement can be achieved if the diffraction fading due to sub-refractive conditions (see time about 2:00) is the most important effect influencing the received power. It suggests that sub-refractive gradients are likely to be approximately the same along the whole propagation path and the approximation of horizontally independent refractivity, which is usually applied in PEM, is reasonable in this case. On the other hand, similar conclusion cannot be reached when multipath propagation occurs because only slight change in a refractivity profile along the propagation path may vary the received power distribution profoundly. These facts have to be kept in mind when the simulation results are interpreted.
Some results of the ongoing studies focussed on the propagation impairments of the atmospheric refractivity in the lowest troposphere were presented. Concurrent measurements of the vertical distribution of atmospheric refractivity together with the multi-receiver microwave propagation experiment were described. A new statistical model of vertical refractivity gradient was introduced. The unique joint statistics of ducting layers parameters were presented. The application of parabolic equation method was demonstrated on the example of a diffraction fading event. Simulated and measured time series were compared. A good agreement between simulation and measured data has been witnessed.
Future works in the area of the atmospheric refractivity related propagation effects should, for example, investigate the relations between the time evolution of duct parameters and multipath propagation characteristics, which is the area where only little is known at this moment.
This research was financially supported by the Czech Science Foundation under the project no. P102/10/1901.
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