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We examine the close relationship between Gaussian processes and the Kalman filter and show how Gaussian processes can be interpreted using familiar Kalman filter mathematical concepts. We use this insight to develop a novel hybrid filter, which we call the KFGP, for spatial-temporal modelling. The KFGP uses Gaussian process kernels to model the spatial field while exploiting efficient Kalman filter state-based approaches to model the temporal component. We also develop a Gaussian process kernel for the familiar Kalman filter near constant acceleration model.