Abstract
Iterative ensemble smoothers (IES) are among the state-of-the-art approaches to solving history matching problems. From an optimization-theoretic point of view, these algorithms can be derived by solving certain stochastic nonlinear-least-squares problems. In a broader picture, history matching is essentially an inverse problem, which is often ill-posed and may not possess a unique solution. To mitigate the ill-posedness, in the course of solving an inverse problem, prior knowledge and domain experience are often incorporated, as a regularization term, into a suitable cost function within a respective optimization problem. Whereas in the inverse theory there is a rich class of inversion algorithms resulting from various choices of regularized cost functions, there are few ensemble data assimilation algorithms (including IES) which in their practical uses are implemented in a form beyond nonlinear-least-squares. This work aims to narrow this noticed gap. Specifically, we consider a class of more generalized cost functions, and establish a unified formula that can be used to construct a corresponding group of novel ensemble data assimilation algorithms, called generalized IES (GIES), in a principled and systematic way. For demonstration, we choose a subset (up to 30 +) of the GIES algorithms derived from the unified formula, and apply them to two history matching problems. Experiment results indicate that many of the tested GIES algorithms exhibit superior performance to that of an original IES developed in a previous work, showcasing the potential benefit of designing new ensemble data assimilation algorithms through the proposed framework.
Highlights
A typical inverse problem involves finding one set(s) of model variables m, as inputs to a numerical forward simulator g, in such a way that the generated outputs g(m) from the simulator can match a collection of observed data do, or other relevant quantities of interest, to a good extent
Norwegian Research Centre (NORCE), 5008 Bergen, Norway instance, it is known as estimation theory in statistical signal processing [1], or data assimilation in meteorology and oceanography [2, 3], or history matching in reservoir engineering [4]
Recent years have witnessed the developments of new algorithms that are able to mitigate some of the noticed computational issues in Bayesian and/or optimization-based inversion approaches, among which we focus here on iterative ensemble smoothers (IES) that are among the state-of-the-art approaches to large-scale history matching problems
Summary
A typical inverse problem involves finding one (or multiple) set(s) of model variables m, as inputs to a numerical forward simulator g, in such a way that the generated outputs g(m) from the simulator can match a collection of observed data do, or other relevant quantities of interest (e.g., a probability density function conditioned on do), to a good extent. Recent years have witnessed the developments of new algorithms that are able to mitigate some of the noticed computational issues in Bayesian and/or optimization-based inversion approaches, among which we focus here on iterative ensemble smoothers (IES) that are among the state-of-the-art approaches to large-scale history matching problems. An open problem for ensemble-based methods is that, except for some special cases, in general the final ensemble of estimated model variables may not be considered as samples drawn from the correct posterior PDF, which satisfies Bayes’ theorem with respect to the corresponding prior PDF and the likelihood function [16]. Experiment results indicate that many of the tested GIES algorithms outperform the original IES in the case studies, manifesting the potential benefit of designing new ensemble data assimilation algorithms under the GMAC framework
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