Multiscale Data Integration Using Markov Random Fields
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AbstractWe propose a hierarchical approach to spatial modeling based on Markov Random Fields (MRF) and multi-resolution algorithms in image analysis. Unlike their geostatistical counterparts that simultaneously specify distributions across the entire field, MRFs are based on a collection of full conditional distributions that rely on local neighborhood of each element. This critical focus on local specification provides several advantages: (a) MRFs are computationally tractable and are ideally suited to simulation-based computation such as MCMC (Markov Chain Monte Carlo) methods, and (b) model extensions to account for non-stationarity, discontinuity and varying spatial properties at various scales of resolution are easily accessible in the MRF framework. Our proposed method is computationally efficient and well suited to reconstruct fine scale spatial fields from coarser, multi-scale samples (e.g., based on seismic and production data) and sparse fine scale conditioning data (e.g., well data). It is easy to implement and can account for the complex, non-linear interactions between different scales as well as precision of the data at various scales in a consistent fashion. We illustrate our method using a variety of examples that demonstrate the power and versatility of the proposed approach. Finally, a comparison with Sequential Gaussian Simulation with Block Kriging (SGSBK) indicates similar performance with less restrictive assumptions.
