Atmospheric Refraction and Propagation in Lower Troposphere (Electromagnetic Waves) Part 1

Introduction

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 topic, recent experimental and modelling results are presented that are related to atmospheric refractivity effects on the propagation of microwaves in the lowest troposphere. The topic 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.

Atmospheric refractivity

Physical parameters of air and refractivity formula

The refractive index of air n is related to the dielectric constants of the gas constituents of an air mixture. Its numerical value is only slightly larger than one. Therefore, a more convenient atmospheric refractivity N (N-units) is usually introduced as:

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It can be simply demonstrated, based on the Debye theory of polar molecules, that refractivity can be calculated from pressure p (hPa) and temperature T (K) as (Brussaard, 1996):

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where e (hPa) stands for a water vapour pressure that is related to the relative humidity H (%) by a relation:

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where es (hPa) is a saturation vapour pressure. The saturation pressure es depends on temperature t (°C) according to the following empirical equation:

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where for the saturation vapour above liquid water a = 6.1121 hPa, b = 17.502 and c = 240.97 °C and above ice a = 6.1115 hPa, b = 22.452 and c = 272.55 °C. 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.

The radio refractivity dependence on temperature and relative humidity of air for pressure p = 1000 hPa (a), refractivity sensitivity dependence on temperature and relative humidity of air (b).

Fig. 1. The radio refractivity dependence on temperature and relative humidity of air for pressure p = 1000 hPa (a), refractivity sensitivity dependence on temperature and relative humidity of air (b).

The sensitivity of refractivity on temperature and relative humidity of air is shown in Fig. 1b. For t = 10°C (cca average near ground temperature in the Czech Republic), H = 70% (cca average near ground relative humidity) and p = 1000 hPa, the sensitivities are dN/dt = 1.43 N-unit/°C, dN/dH = 0.57 N-unit/% and dN/dp = 0.27 N-unit/hPa. The refractivity variation is usually most significantly influenced by the changes of relative humidity as a water vapour content often changes rapidly (both in space and time) and it is least sensitive to pressure variation. However a decrease in pressure with altitude is mainly responsible for a standard vertical gradient of the atmospheric refractivity. During standard atmospheric conditions, the temperature and pressure are decreasing with the height above the ground with lapse rates of about 6 °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 h can be obtained using pressure and temperature sensitivities and their standard lapse rates. Such an estimated standard vertical gradient of refractivity is about dN/dh » -42 N-units/km. It will be seen that such value is very close to the observed long term median of the vertical gradient of refractivity.

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:

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where a position vector r is associated with each point along a ray and s is the curvilinear abscissa along this ray. Since the atmosphere is dominantly horizontally stratified, the gradient Vn has its main component in vertical direction. Considering nearly horizontal propagation, the refractive index close to one and only vertical component of the gradient Vn, one can derive from (5) that the inverse of the radius of ray curvature, p, is approximately equal to the negative height derivative of the refractive index, -dn/dh. Using the conservation of a relative curvature: 1/R – 1/p = const. = 1/Ref – 1/<» one can transform the curvilinear ray to a straight line propagating above an Earth surface with the effective Earth radius Ref given by:

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where R stands for the Earth radius and dN/dh denotes a vertical gradient of refractivity. Three typical propagation conditions are observed depending on the numerical value of the gradient. If dN/dh » -40 N-units/km, than from (6): Ref» 4/3 R and standard atmospheric conditions take place. The standard value of the vertical refractivity gradient is approximately equal to the long term median of the gradient observed in mild climate areas. The median gradients observed in other climate regions may be slightly different, see the world maps of refractivity statistics in (Rec. ITU-R P.453-9, 2009).

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 Ref decreases, terrain obstacles are relatively higher and the received signal may by attenuated due to diffraction loss appearing if the obstacle interfere more than 60% of the radius of the 1st Fresnel ellipsoid on the line between the transmitter and receiver. During super-refractive conditions, on the other hand, the effective Earth radius is lower than the Earth radius R or it is even negative when dN/dh < -157 N-units/km. It means a radio path is more "open" in the sense that terrain obstacles are relatively lower. Super-refractive conditions are often associated with multipath propagation when the received signal fluctuates due to constructive and destructive interference of EM waves coming to the receiver antenna with different phase shifts or time delays.

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 statistics of atmospheric refractivity and related propagation effects are of main interest. The statistical data important for the design of terrestrial radio systems have to be obtained from the experiments, an example of which is described further.

Measurement of refractivity and propagation

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.

(a) The terrain profile of an experimental microwave path, TV Tower Prague -Podebrady mast, with the first Fresnel ellipsoids of the lowest and the highest paths for k = Ref/R = 4/3, (b) the parabolic receiver antennas placed on the 150 m high mast (Podebrady site).

Fig. 2. (a) The terrain profile of an experimental microwave path, TV Tower Prague -Podebrady mast, with the first Fresnel ellipsoids of the lowest and the highest paths for k = Ref/R = 4/3, (b) the parabolic receiver antennas placed on the 150 m high mast (Podebrady site).

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 (k = Ref/R = 4/3) the lowest path is attenuated due to the diffraction loss of about 6 dB. Tables 1a and 1b show the parameters of the measurement setup.

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

Table 1a. The parameters of a measurement system (meteorology).

TX tower ground altitude

258.4 m above sea level

TX antenna height

126.3 m

Frequency

10.671 GHz

Polarization

Horizontal

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

Table 1b. The parameters of a measurement system (radio, TX = transmitter, RX = receiver).

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 M is calculated from refractivity N as:

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where h(km) stands for the height above the ground. The reason of using M instead of N here is to clearly point out the possible ducting conditions (dN/dh < -157 N-units/km) when dM/dh < 0 M-units/km.

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 dM/dh near the ground between 2:00 and 4:00 which is caused by strong temperature inversion together with no compensating humidity effect. The received signal measured on the higher antennas that are not affected by diffraction stays around the nominal value with some smaller fluctuations probably due to multipath and focusing/defocusing effects. A typical example of multipath propagation is shown in Fig. 6. In the middle of the day from about 7:00 to 18:00, the received signal is steady at all heights and the atmosphere seems to be well mixed. On the other hand, multipath propagation occurring in the morning and at night is characterized by relatively fast fluctuations of the received signal. It is seen that all the receivers are impaired in the particular multipath events. Deep fading (attenuation > 20 dB) is quite regularly changing place with significant enhancement of the received signal level.

The vertical profiles of temperature T, relative humidity H, modified refractivity M and received signal levels relative to free-space level observed on 17 November 2010

Fig. 3. The vertical profiles of temperature T, relative humidity H, modified refractivity M and received signal levels relative to free-space level observed on 17 November 2010

The vertical profiles of temperature T, relative humidity H, modified refractivity M and received signal levels relative to free-space level observed on 26 June 2010

Fig. 4. The vertical profiles of temperature T, relative humidity H, modified refractivity M and received signal levels relative to free-space level observed on 26 June 2010

The vertical profiles of temperature T, relative humidity H, modified refractivity M and received signal levels relative to free-space level observed on 14 October 2010

Fig. 5. The vertical profiles of temperature T, relative humidity H, modified refractivity M and received signal levels relative to free-space level observed on 14 October 2010

The vertical profiles of temperature T, relative humidity H, modified refractivity M and received signal levels relative to free-space level observed on 12 September 2010

Fig. 6. The vertical profiles of temperature T, relative humidity H, modified refractivity M and received signal levels relative to free-space level observed on 12 September 2010

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