Ensemble Kalman methods for high-dimensional hierarchical dynamic space-time models

We propose a new class of filtering and smoothing methods for inference in high-dimensional, nonlinear, non-Gaussian, spatio-temporal state-space models. The main idea is to combine the ensemble Kalman filter and smoother, developed in the geophysics literature, with state-space algorithms from the statistics literature. Our algorithms address a variety of estimation scenarios, including on-line and off-line state and parameter estimation. We take a Bayesian perspective, for which the goal is to generate samples from the joint posterior distribution of states and parameters. The key benefit of our approach is the use of ensemble Kalman methods for dimension reduction, which allows inference for high-dimensional state vectors. We compare our methods to existing ones, including ensemble Kalman filters, particle filters, and particle MCMC. Using a real data example of cloud motion and data simulated under a number of nonlinear and non-Gaussian scenarios, we show that our approaches outperform these existing methods

Particle Learning and Smoothing

Particle learning (PL) provides state filtering, sequential parameter learning and smoothing in a general class of state space models. Our approach extends existing particle methods by incorporating the estimation of static parameters via a fully-adapted filter that utilizes conditional sufficient statistics for parameters and/or states as particles. State smoothing in the presence of parameter uncertainty is also solved as a by-product of PL. In a number of examples, we show that PL outperforms existing particle filtering alternatives and proves to be a competitor to MCMC.Comment: Published in at http://dx.doi.org/10.1214/10-STS325 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

A Bayesian Consistent Dual Ensemble Kalman Filter for State-Parameter Estimation in Subsurface Hydrology

Ensemble Kalman filtering (EnKF) is an efficient approach to addressing uncertainties in subsurface groundwater models. The EnKF sequentially integrates field data into simulation models to obtain a better characterization of the model's state and parameters. These are generally estimated following joint and dual filtering strategies, in which, at each assimilation cycle, a forecast step by the model is followed by an update step with incoming observations. The Joint-EnKF directly updates the augmented state-parameter vector while the Dual-EnKF employs two separate filters, first estimating the parameters and then estimating the state based on the updated parameters. In this paper, we reverse the order of the forecast-update steps following the one-step-ahead (OSA) smoothing formulation of the Bayesian filtering problem, based on which we propose a new dual EnKF scheme, the Dual-EnKF$_{\rm OSA}$. Compared to the Dual-EnKF, this introduces a new update step to the state in a fully consistent Bayesian framework, which is shown to enhance the performance of the dual filtering approach without any significant increase in the computational cost. Numerical experiments are conducted with a two-dimensional synthetic groundwater aquifer model to assess the performance and robustness of the proposed Dual-EnKF$_{\rm OSA}$, and to evaluate its results against those of the Joint- and Dual-EnKFs. The proposed scheme is able to successfully recover both the hydraulic head and the aquifer conductivity, further providing reliable estimates of their uncertainties. Compared with the standard Joint- and Dual-EnKFs, the proposed scheme is found more robust to different assimilation settings, such as the spatial and temporal distribution of the observations, and the level of noise in the data. Based on our experimental setups, it yields up to 25% more accurate state and parameters estimates