Precipitation is a complex physical process that varies in space and time. Predictions and interpolations at unobserved times and/or locations help to solve important problems in many areas. In this paper, we present a hierarchical Bayesian model for spatio-temporal data and apply it to obtain short term predictions of rainfall. The model incorporates physical knowledge about the underlying processes that determine rainfall, such as advection, diffusion and convection. It is based on a temporal autoregressive convolution with spatially colored and temporally white innovations. By linking the advection parameter of the convolution kernel to an external wind vector, the model is temporally nonstationary. Further, it allows for nonseparable and anisotropic covariance structures. With the help of the Voronoi tessellation, we construct a natural parametrization, that is, space as well as time resolution consistent, for data lying on irregular grid points. In the application, the statistical model combines forecasts of three other meteorological variables obtained from a numerical weather prediction model with past precipitation observations. The model is then used to predict three-hourly precipitation over 24 hours. It performs better than a separable, stationary and isotropic version, and it performs comparably to a deterministic numerical weather prediction model for precipitation and has the advantage that it quantifies prediction uncertainty.