Abstract: |
To the statistically minded, the subsurface presents a complex, spatially correlated "mixture" distribution of rock properties to our remote sensing tools, where the mixture originates from different rock types. In petroleum contexts, the information we collect from seismic, EM and production data is often dominated by the geometrical boundaries separating lithologies in the subsurface, yet many standard geophysical inversion tools use purely continuous optimization techniques that model rock properties as if they come from some common population. This pooling approximation imposes a strong prior-model footprint on inversion results. Newer hierarchical Bayesian approaches that embed a discrete aspect via discrete Markov random fields, coupled with conditional prior distributions that embed rock-physics relationships, offer a tractable way to represent the categorical aspects of geology. Some published studies on these models, using seismic data, indicate the posterior distribution can be modestly sharp, though the sampling MCMC algorithms are naturally challenging.
This renews interest in the maximum aposteriori model. We present some theory and examples using algorithms drawn from the computer-vision literature for maximum aposteriori model inference in joint lithology-fluid/rock-properties problems using seismic. The expectation-maximisation algorithm is the natural framework to use, and the main challenge is to obtain good approximations to the marginal distributions required for the expectation step in very high dimensional problems. Recent work in belief-propagation techniques has made these MRF-based marginalisation and maximisation problems much more computationally feasible, and the principle tools are the iterative sum-product and max-product algorithms. Faster but "hard" classifications are also available by graph-cutting techniques. A secondary challenge is performing maximisation in a spatial and multivariate coupled setting, for which purpose continuous Gaussian Markov random fields are a natural candidate. An intuitively pleasant result is that the rock--properties inference can be cast as a facies-membership reweighted-prior least-squares problem, which can be tackled by the usual conjugate gradient apparatus. |
