approach and its inherent lack of precision to some degree, however, various other attempts have been made to predict point rain-rate distributions, or statistics on such distributions, at specific points within a region. Such approaches have normally used measured rain-rate distributions for as many locations in a region as possible, and then fitted contour maps for particular parameters of the distribution. Perhaps the first successful approach of this type was that of Segal for Canada (Segal B, 1986), which employed contour maps of two parameters of the rain rate distribution, based on data for 47 locations within Canada and adjacent regions of the United States. Watson et al.
(1982) later mapped rain rates exceeded for 0.1% and 0.01% of an average year based on data for 400 locations within Europe. Moupfouma (1987) developed two more general global models using two parameters, one of which was conveniently the rain rate exceeded for 0.01% of the time.
Another attempt to avoid the zonal approach was made by Rice and Holmberg (1973), who developed a model employing three basic long-term parameters: the average annual rainfall (mm), the ratio of thunderstorm rain to total rain, and the average annual number of days with rainfall of 0.25 mm or more. The ITU-R (and its predecessor, the CCIR) has provided a global contour map of the rain rate exceeded for 0.01% of the year since 1982 ITU-R P.837-1 (1994), although it was derived in part from use of the zonal model, at least at the early stages. The accuracy of the global map of the ITU-R has of course always been quite variable because of the lack of data for several regions of world including Africa. This study therefore partly attempts to address this challenge.
conversion models into the three categories of physical, analytical and empirical models.
A proposed hybrid method for the conversion of five-minute integration time to one- minute equivalent was discussed as well as comparison of the proposed model with two suggested global models.
In the proposed hybrid method, the strength of each category of model classes were combined to produced a hybrid model. The selected model consists of regional parameters in order to characterize the rain rate pattern for a defined area. The resulting one-minute cumulative distribution of rain rate is fitted with polynomial, power, linear and logarithm fits of distributions. The performances of these fits were optimized using standard deviation and root-mean square of absolute relative error at the control site of Durban. The optimizations were carried out over the equiprobable approach and conversion factor approach. The results show that the equiprobable approach gives better results than the conversion factor approach. In addition, the second order polynomial fit performed relative better than its counterparts. The evidence of good performance by the second order polynomial fit makes it a good candidate for conversion of rain rate from five-minute to equivalent one-minute in South Africa and the surrounding Islands. The South African region was subsequently classified into 12 sub-climatic zones using the Köppen climatic classification. The coefficients of polynomial, power, linear and logarithm fits were given for each of the classes with their square correlation coefficients.
Finally, based on the work in this chapter, the polynomial fit of second order is adopted for South Africa and the two surrounding Islands.
6.1.2 Chapter 4 – Rain Rate Modelling
In this Chapter, the most widely used rain rate probability distributions were investigated using the maximum likelihood estimator to optimize the distributions. It was found that most of the studied areas were best defined by the Gamma distribution model, followed by Weibull distribution model, while the lognormal distribution did well at only a few sites. For the 21 stations, the Gamma model had an averageχ2 statistic of 10.8, followed by the Weibull model with an average of 19.85. The rest of the models gave an average
χ2 statistic of above 55, which is too close to the threshold 63.7 for 40 degrees of freedom. The average root mean square percentage also gives credence to the fact that the Gamma model is the most appropriate to describe most sites in South Africa and surrounding islands. The proposed model will thus serve as a good tool for system designers because it is easy to use when rainfall distribution is needed for any site, and its parameters are easily understood.
The study proposes new rainfall climatic zones based on the ITU-R and the Crane rain climatic zone designations. It is found in several sites that the ITU-R model over- estimated, while in many others the ITU-R model under-estimated, rainfall rate values at defined probabilities of exceedence. It is observed that the widest percentage of differences between ITU-R P.837-1 and P.837-5 was 66.67% for Marion Island, and the least percentage was 13.63% observed in Beaufort and Cape Point. Crane rain climatic zone designations matches in some sites as noted in Pretoria, Tshipise, Cape Town, Beaufort, Klerksdorp and Upington.
Rain contour maps have been identified as desirable tools for the objective of providing system designers, site engineers and network planners with estimated fade margins due to rain attenuation. The two plotted contour maps were tailored to satisfy accepted rainfall climatic zones defined by the ITU-R and Crane maps which are available in most radio planning tools. In this research, the contour map was developed using advanced Geographic Information Systems (GIS) tools with the adoption of IDW estimator to provide the contour map for rainfall rate at 0.01% of exceedence.
6.1.3 Chapter 5 – Rain Dropsize Distribution Modelling
The purpose of this chapter was to model the characteristics of raindrop size distribution for the Southern Africa region. The importance of DSD modelling arises from the need to employ Mie scattering theory and appropriate, region-dependent DSD’s (not just the Laws and Parsons (LP) drop size distribution) to obtain plots of specific attenuation γ (dB/km) against frequency for varying rain rate. The raindrop size distribution spectra showed that the major part of the drop density falls within the 0.3 mm and 4.0 mm
diameter range. Using the maximum likelihood regression method on the collected data, the three-parameter lognormal distributions are estimated for two rain rate regimes. The technique employed gives proper description of the DSD distribution curves very well even at lower rain rates. In this study, it was confirmed that Marshall and Palmer and other considered DSD models are not adequate to describe raindrop size distributions in the Southern Africa region. A simple lognormal distribution model is thus developed to describe the raindrop distribution for Durban using two rain rate regimes. Based on the comparisons carried out, it was found that the proposed model performs better than its counterparts, though at higher rain rates, the distribution patterns of Timothy et al. (2002) and Ajayi et al. (1985) seem to give similar shapes of distribution.