Introduction
1.5 Outline of the dissertation
The numerical calculation of the scattering and attenuation of electromagnetic waves by rain requires detailed knowledge of the microphysical properties of raindrops such as raindrop size, shape, fall velocity and drop-size distribution (OSD). This is the focus of Chapter 2 of this dissertation.
In Section 2.2, the fall velocity of raindrops is examined using the Gunn and Kinzer (1949) measurements of the terminal velocity at sea level and several analytical descriptions thereof are discussed. Thereafter models for arbitrary atmospheric conditions are examined, concluding with the Best (1950) model for variation with height in the S.T. (Summer Tropical) and LC.A.N. standard atmospheres (International Committee of Air Navigation).
The size and shape of a raindrop is necessary for the calculation of the scattering and extinction cross section of a single raindrop. A brief preamble to raindrop scattering is given in Section 2.3. To extend the scattering and extinction model of a single raindrop to that experienced during a rain event, the drop-size distribution of raindrops is required. OSOs exhibit significant spatial and temporal variability. The average OSOs for several rainfall types are thus examined in Section 2.4. Firstly, the Laws and Parson (1943) measurements of the drop-size distribution is discussed. Thereafter several
analytical descriptions for the DSD are investigated including the exponential, gamma, lognormal and Weibull functions.
This chapter is concluded with an investigation of the dielectric properties of water in Section 2.5. The Debye approximations and empirical models of Ray (1972) and Liebe (1991) are examined for frequencies up to 1 THz. The frequency and temperature dependent complex refractive index of water is instrumental in the calculation of the total and scattering cross-section of raindrops.
The attenuation and scattering of electromagnetic waves by raindrops is evaluated by applying the scattering theory for a single, lossy dielectric sphere. Typical raindrops have diameters ranging from 0.1 mm to 7 mm. Thus, for millimetre-wave frequency range the drop-size is comparable to wavelength hence Mie scattering theory is applicable. Evaluation begins with the computation of the extinction and scattering cross sections of a single raindrop as examined in Section 3.2.
The specific attenuation is heavily dependent upon the chosen DSD and statistics relating to the location of these drops within the volume. In this study it is assumed that the raindrops are distributed throughout the volume in accordance with the Poisson process. In most calculations the Laws and Parsons DSDs is typically assumed. In Section 3.4 several DSD models have been evaluated from the conventional exponential distributions to the gamma, lognormal and Weibull distributions. The more sophisticated DSDs overcome the shortfalls of the exponential distributions which tend to over-estimate the number of small and large raindrops.
In section 3.5 the difference between the extinction cross section is evaluated for spherical raindrops and for the more realistic Prupacher and Pitter (1971). Vertical polarizations will have a slightly smaller specific attenuation and horizontal polarizations a slightly larger specific attenuation than that calculated using Mie theory
for spheres. Hence future work should entail the evaluation of the extinction cross sections for realistically-shaped raindrops and include the effect of drop shape in this model.
The final step that remains is the prediction of the rain attenuation statistics. To achieve this short-duration precipitation rates are required. This is the focus of Chapter 4. First, the global rain-rate climate models are discussed in Section 4.2. These models are based upon the climatic classification of the region, thus a description of the various climatic zones in South Africa is also useful and is provided in the Appendix. Both Crane's global rain-rate climate model and the ITU-R P.837 global rain-rate model are discussed and simulated.
Thereafter a discussion of extreme value theory and the procedure for obtaining 5- minute rain-rate distributions for any location in South Africa is given in Section 4.3.
The procedure involves the use of the depth-duration-frequency curve for South Africa.
More accurate 5-minute distributions can be provided by using actual rainfall data from annual maxima series when available. Consequently 5-minute distributions are obtained for 8 locations with approximately 40 years of measurements. Thereafter these distributions are converted to I-minute rain-rate distributions using the conversion ratio of Ajayi and Ofoche (1983).
Other models such as those of Moupfouma (1987) and Moupfouma and Martin (1995) are also discussed and simulated for South Africa in Section 3.4 and Section 4.5, respectively. A comparison of all the models with I-minute rain-rate measurements in Durban is given in Section 4.6.
This estimate of the surface-point I-minute rain rate is only the first step. Rain exhibits significant spatial and temporal variation. Hence the rain-rate profile along the
propagation path is also required. The Crane (1980) attenuation model and the ITU recommendation (ITU-R P530-9) are discussed in Section 4.7.
In Chapter 5, the path attenuation and exceedance probabilities are evaluated for the 6.73 km LOS link established between the two campuses of the University of KwaZulu- Natal (UKZN). The various LOS link details such as the equipment setup, link parameters, path profile and link budget were provided in Section 5.2. Sample link measurements are also given.
In Section 5.3 the specific attenuation was calculated using the exponential, gamma, lognormal and Weibull DSDs. Analysis of the effects of the DSD on the specific attenuation is then discussed.
The Crane (1980, 1996) attenuation model is then applied to convert the surface-point rain-rate to a path integrated rain-rate. The results are given in Section 5.4. The results in this chapter focus on 19.5 GHz, the operating frequency of the LOS link. At the frequencies around 19.5 GHz, the specific attenuation and hence the path attenuation is relatively invariant of the DSD. This is because at 19.5 GHz the attenuation is strongly influenced by the medium-sized raindrops. At others frequencies, the variations due to the DSD are quite considerable and can result in changes up to 50 % in the calculated path attenuation. To illustrate this effect, analysis was also performed at the frequencies
f =
4, 12, 15,40 and 80 GHz and the results are given in the Appendix D.Finally, the various I-minute rain-rate distributions discussed in Chapter 4 were applied to provide the exceedance probabilities for the path attenuation on the LOS link. The results using the proposed extreme-value model for the I-minute cumulative distributions, along with the Seeber (1985) model, the ITU-R P.837 global rain-rate climate model, the Moupfouma (1987) model and the Moupfouma and Martin (1995) temperate and tropical/subtropical models are presented in Section 5.5. These
cumulative rain-rate distributions together with the path attenuation can be used to analyse the time any particular path attenuation is exceeded.
Finally the results presented in this dissertation are summarised along with concluding remarks in Chapter 6. Future work in the field is also provided. Knowledge of the specific attenuation is invaluable to determine the path lengths, operating frequency and fade margins required for communication systems. Since the specific attenuation is heavily dependent on the DSD, the influence of the DSD on link availability must be examined and hence is the subject of this research.
During rain events, a large fade margin may often be required. It would be uneconomical to implement such margins since they are only needed for a fraction of the time. A cumulative distribution model of the fade depth and duration due to rain is thus a valuable tool for a system designer. With such models the system designer can then determine the appropriate fade margin and resulting outage period for the link.