Abstract: |
The Bayesian framework provides a consistent scientific framework to model uncertainty of complex spatial inverse problems. To generate a posterior set of models, two approaches are currently used: sampling methods such as Markov Chain Monte-Carlo (McMC) and optimization methods. Optimization methods (gradients, EnKf) are often employed because they are relatively fast. However, they result in a reduction of uncertainty that is not consistent with Bayes? rule, or with the given prior uncertainty and likelihood distribution. Sampling methods are consistent with Bayes? rule, and are therefore preferable compared to optimization. However, sampling methods are often infeasible when the forward model is too CPU demanding. In this work, we propose to model uncertainty using a sampling technique which is geologically consistent and adheres to Bayes? rule, yet need not rely on the traditional McMC framework. The method relies on a distance-based kernel K-L expansion which is deduced from the initial set of Earth models sampled from the prior. Efficiency is obtained in two ways: (1) the introduction of a metric space makes the parameterization data or forward model-dependant; (2) the sampling is performed on a low dimensional Gaussian vector which is derived from the expansion. Finally, a pre-image problem is solved using the Extended Probability Perturbation Method in order to create models corresponding to the sampled Gaussian vector and to ensure consistency with the prior model statistics. We present a real 3D case where the prior model is modeled using a higher-order spatial model and the forward model is a flow simulator. We show how the number of forward model runs can be significantly reduced compared to rejection sampling, while maintaining similar posterior uncertainty. |
