Deterministic Linear Bayesian Updating of State and Model Parameters

Bayesian estimation has become an important topic for inverse problems in the context of hydrocarbon recovery. The conceptual and computational advantages due to direct integration with uncertainty quantification workflows are appealing. Especially, linear Bayesian techniques like the ensemble Kalman filter (EnKF) have been successfully used in numerous cases. However, such techniques have difficulties in some applications which are often caused by sampling errors, a limited ensemble size, or the sometimes large number of required samples. In this work we present and discuss a closely related linear Bayesian technique which is based on orthogonal expansions of the stochastic spectrum of the involved random variables and random fields. Basically being a family of fully deterministic implementations of the well-known projection theorem of Hilbert spaces, the technique is conceptually simple, yet powerful. Since they are fully deterministic, these methods avoid all sampling errors. First combined parameter and state estimation results with a low-dimensional chaotic model are presented, using a specific choice of orthogonal expansion. These are compared to results obtained with EnSRF, since it is a close relative to these spectral estimation methods. Challenges and opportunities for applications to the inverse problem of identification for hydrocarbon reservoirs are discussed.