Simulations based on a month of radar data from Florida, and a summer of radar data from Nelspruit, South Africa, were used to quantify the errors in the measurement of mean areal rainfall which arise simply as a result of the extreme variability of convective rainfall, even with perfect remote sensing instruments. The raingauge network measurement errors were established for random and regular network configurations using daily and monthly radar-rainfall accumulations over large areas. A relationship to predict the measurement error for mean areal rainfall using sparse networks as a function of raining area, number of gauges, and the variability of the rainfield was developed and tested. The manner in which the rainfield probability distribution is transformed under increasing spatial and temporal averaging was investigated from two perspectives. Firstly, an empirical relationship was developed to transform the probability distribution based on some measurement scale, into a distribution based on a standard measurement length. Secondly, a conceptual model based on multiplicative cascades was used to derive a scale independent probability distribution.