  We construct flexible spatio-temporal models through stochastic partial
differential equations (SPDEs) where both diffusion and advection can be
spatially varying. Computations are done through a Gaussian Markov random field
approximation of the solution of the SPDE, which is constructed through a
finite volume method. The new flexible non-separable model is compared to a
flexible separable model both for reconstruction and forecasting and evaluated
in terms of root mean square errors and continuous rank probability scores. A
simulation study demonstrates that the non-separable model performs better when
the data is simulated with non-separable effects such as diffusion and
advection. Further, we estimate surrogate models for emulating the output of a
ocean model in Trondheimsfjorden, Norway, and simulate observations of
autonoumous underwater vehicles. The results show that the flexible
non-separable model outperforms the flexible separable model for real-time
prediction of unobserved locations.
