The ensemble Kalman filter (EnKF) is a sequential Monte Carlo method for solving nonlinear spatiotemporal inverse problems, such as petroleum-reservoir evaluation, in high dimensions. Although the EnKF has seen successful applications in numerous areas, the classical EnKF algorithm can severely underestimate the prediction uncertainty. This can lead to biased production forecasts and an ensemble collapsing into a single realization.
In this paper, we combine a previously suggested EnKF scheme based on dimension reduction in the data space, with an automatic cross-validation (CV) scheme to select the subspace dimension. The properties of both the dimension reduction and the CV scheme are well known in the statistical literature. In an EnKF setting, the former can reduce the effects caused by collinear ensemble members, while the latter can guard against model overfitting by evaluating the predictive capabilities of the EnKF scheme. The model-selection criterion traditionally used for determining the subspace dimension, on the other hand, does not take the predictive power of the EnKF scheme into account, and can potentially lead to severe problems of model overfitting. A reservoir case study is used to demonstrate that the CV scheme can substantially improve the reservoir predictions with associated uncertainty estimates.