Presented is a rainfall prediction methodology for application in operational hydrologic forecasting with forecast lead times of 1--6 hours and spatial-resolution scales of 10--30 km. The essential elements of the prediction methodology are a mathematical model for precipitation prediction from surface and upper air meteorological variables; operational forecasts of temperature, pressure, humidity, and wind fields by large-scale numerical weather prediction models; surface and upper air meteorological observations; remote and on-site rainfall observations; and a state estimator for real-time updating from local frequent rainfall observations and for probabilistic predictions. This paper formulates a class of rainfall models suitable for this prediction methodology. The models are based on the differential equation of conservation of cloud and rainwater equivalent mass and on a newly introduced advection equation for a parameter that determines updraft strength. The latter advection equation is a prognostic equation for the strength of convection in space and time. The innovative features of the model formulated and tested are the inclusion of the prognostic equation for the advection of regions of active convection, the formulation of the state estimator component for state updating and probabilistic forecasts, and the utilization of a numerical solution scheme which reduces artificial numerical diffusion and can be used with the state estimator because of its explicit form. Utilization of the prediction model formulated was exemplified in several case studies of summer convection in Oklahoma using data available during routine forecast operations. The case studies show that when verified with radar rainfall data, the model's hourly precipitation predictions over a 20,000 km2 area with a 100--900 km2 resolution are better than simple persistence and explain more than 60% of the observed hourly rainfall variance. Sensitivity studies quantify dependence of rainfall predictions to microphysical and state-estimator parameters. ¿ American Geophysical Union 1996