Particle filters are fully nonlinear data assimilation methods and as such are highly relevant. While the standard particle filter degenerates for high‐dimensional systems, recent developments have opened the way for new particle filters that can be used in such systems. The implicit equal‐weights particle filter (IEWPF) is an efficient approach that avoids filter degeneracy because it gives equal particle weights by construction. The method uses implicit sampling, whereby auxiliary vectors drawn from a proposal distribution undergo a transformation before they are added to each particle. In the original formulation of the IEWPF, the proposal distribution has a gap, causing all but one particle to have an inaccessible region in state space. We show that this leads to a systematic bias in the predictions and we modify the proposal distribution to eliminate the gap. We achieved this by using a two‐stage proposal method, where a single variance parameter is tuned to obtain adequate statistical coverage properties of the predictive distribution. We discuss the properties of the implicit mapping from an auxiliary random vector to the state vector, keeping in mind the aim of avoiding particle resampling. The revised filter is tested on linear and weakly nonlinear dynamical models in low‐dimensional and moderately high‐dimensional settings, demonstrating the success of the new methodology in removing the bias.

We consider the problem of conditioning a geological process-based computer simulation, which produces basin models by simulating transport and deposition of sediments, to data. Emphasising uncertainty quantification, we frame this as a Bayesian inverse problem, and propose to characterize the posterior probability distribution of the geological quantities of interest by using a variant of the ensemble Kalman filter, an estimation method which linearly and sequentially conditions realisations of the system state to data.A test case involving synthetic data is used to assess the performance of the proposed estimation method, and to compare it with similar approaches. We further apply the method to a more realistic test case, involving real well data from the Colville foreland basin, North Slope, Alaska.

Several applications rely on data assimilation methods for complex spatio-temporal problems. The focus of this paper is on ensemble-based methods, where some approaches require estimation of covariances between state variables and observations in the as-similation step. Spurious correlations present a challenge in such cases as they can degrade the quality of the ensemble represen-tation of probability distributions. In particular, prediction vari-ability is often underestimated. We propose to replace the sample covariance estimate by a parametric approach using maximum likelihood estimation for a small number of parameters in a spatial covariance model. Parametric covariance and precision estimation are employed in the context of the ensemble Kalman filter, and applied to a Gauss-linear autoregressive model and a geological process model. We learn that parametric approaches reduce the underestimation in prediction variability. Furthermore rich, non-stationary models do not seem to add much over simpler models with fewer parameters.

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