We introduce a class of multi-scale models for random fields. The novel framework couples standard Markov models for the random field stochastic process at different levels of resolution, and links them via error models to induce a new and rich class of structured linear models reconciling modelling and information at different levels of resolution. Jeffrey's rule of conditioning is used to revise the implied distributions and ensure that the probability distributions at different levels are strictly compatible. Bayesian estimation based on Markov Chain Monte Carlo methods is developed. To highlight the potential applications of our multi-scale framework, we provide two examples. In the first example, we illustrate with a simulated data set the procedures of multi-scale field simulation and parameter estimation. In the second example, we use our multi-scale model as a prior for permeability fields to solve a fluid flow inverse problem.

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