A stochastic space-time model of rainfall is proposed, which is defined by a small number of parameters and models rainfall intensity images measured by radar. It is a phenomenological (rather than a physically based) model which provides sequences of realistic stochastically generated rainfields over an area covered by an S-band weather radar sited near Bethlehem, South Africa. The weather types studied vary from predominantly scattered convective thunder-shower type rainfall through to widespread, general, stratiform rainfields with high Wetted Area Ratios. The methods of analysis and modelling of rainfall events as measured by radar are described. A three dimensional (two space and one time) simulated rainfall event designed to mimic a real sequence is generated, analysed and compared to the observed rainfall event. Not only does it reproduce the statistics without obvious bias, it has been validated using the Generalized structure function. Variograms (specialisations of the Generalized structure function) fitted to the data measured by radar from a variety of rainfield types suggest that the correlation distance in rainfields is limited to between 12 and 25 km. The implication is that although the rainfields appear to be locally non-stationary, they are stationary over areas typically measured by radar. This is a post-justification for using methods based on the assumption that rainfields are stationary random fields and allows the use of straightforward statistical and spectral techniques which are the heart of the String of Beads model. This paper concentrates on modelling the rainfall process during a wet period, deferring the treatment of the inter-arrival of wet and dry periods to a follow-on paper.