An extreme-value model of surface-point rain-rate distribution for the prediction of microwave rain attenuation in Southern Africa

Abstract

A semi-empirical model for the probability distribution of point rainfall rate is presented. The model was developed for predicting the probability distribution of rain attenuation on microwave links on the Southern African subcontinent. The attenuation distribution is required for obtaining the average annual outage of terrestrial and earth-to-satellite communication links. The model is an extension of the extreme-value method presented by Lin. As in Lin's method the distribution of randomly varying rain rate is approximated by the lognormal distribution and the distribution of annual rain-rate maxima by the log-Gumbel distribution. The extension of the extreme-value method comprises new, generalized relationships for the scale parameter a and the position parameter U of the log-Gumbel distribution. The expressions developed for U are a function of mean annual rainfall (MAR), climatic region and rain-gauge integration time, while a has a constant value. The relationships were obtained from an analysis of the depth-duration-frequency diagram for Southern Africa, by Midgley and Pitman. The resulting rain-rate model gives distributions which are a function of only MAR and climatic region ('coastal' or 'inland'). Integration time is variable, ranging from 5 to 1440 minutes. Curves of the cumulative rain-rate distribution are given for various integration times, for various mean annual rainfalls and for the two climatic regions concerned. By using the curves the rain-rate distribution and -- by means of existing attenuation models -- also the rain-attenuation distribution, may be obtained for locations for which no rain-rate data are available. The 15-minute rain-rate predictions for locations on the Southern African subcontinent agree well with observed distributions. However, for two islands in the South Indian and South Atlantic oceans, respectively, the results were poor. However, this does not necessarily invalidate the extreme-value theory used in the model. The model gives rain-rate estimates for the Southern African subcontinent which are equivalent to a two-year measurement programme. Although the model was developed to provide distributions of primarily a 5-min integration time, a lack of data precluded testing the model at this integration time for Southern Africa. However, the model was extensively tested at 5-min integration time for USA locations, with satisfactory results. The model was also tested for predicting the rain-attenuation distribution on earth-to-satellite microwave links in the USA, using a well-known attenuation model. These results were very satisfactory. It is concluded that the model should be suitable for predicting rain attenuation on the Southern African subcontinent, not only for earth-to-satellite links, but also for terrestrial links. The model obviates geographical interpolation of rain-rate and rain-attenuation distributions. The results for Southern Africa and the USA support the log-Gumbel and lognormal approximations, as well as Lin's extreme-value theory. That MAR is an important determinant of the rain-rate distribution is in agreement with the Rice-Holmberg model. The results are furthermore in agreement with the hypothesis that the spatial and temporal characteristics of rain are relatively invariant within large geographical regions, while the frequency of rain occurrences varies from location to location. The approach followed in developing the model could be used for establishing models of greater geographical resolution and extent.