Blur-generated non-separable space-time models

Statistical space-time modelling has traditionally been concerned with sepa rable covariance functions, meaning that the covariance function is a produ ct of a purely temporal function and a purely spatial function. We draw att ention to a physical dispersion model which could model phenomena such as t he spread of an air pollutant. We show that this model has a non-separable covariance function. The model is well suited to a wide range of realistic problems which will be poorly fitted by separable models. The model operate s successively in time: the spatial field at time t + 1 is obtained by 'blu rring' the field at time t and adding a spatial random field. The model is first introduced at discrete time steps, and the limit is taken as the leng th of the time steps goes to 0. This gives a consistent continuous model wi th parameters that are interpretable in continuous space and independent of sampling intervals. Under certain conditions the blurring must be a Gaussi an smoothing kernel. We also show that the model is generated by a stochast ic differential equation which has been studied by several researchers prev iously.