Ensemble transform Kalman-Bucy filters

TL;DR

This paper introduces stable, computationally efficient ensemble transform Kalman-Bucy filters, analyzing their behavior and demonstrating promising results across multiple models, including Lorenz systems and an atmospheric model.

Contribution

It develops transform-based ensemble Kalman-Bucy filters with stable integration schemes and evaluates their performance on diverse models, advancing data assimilation methods.

Findings

The proposed schemes are stable and computationally efficient.

They perform well on Lorenz models and the SPEEDY atmospheric model.

Encouraging results suggest further research is warranted.

Abstract

Two recent works have adapted the Kalman-Bucy filter into an ensemble setting. In the first formulation, BR10, the full ensemble is updated in the analysis step as the solution of single set of ODEs in pseudo-BGR09, the ensemble of perturbations is updated by the solution of an ordinary differential equation (ODE) in pseudo-time, while the mean is updated as in the standard KF. In the second formulation, BR10, the full ensemble is updated in the analysis step as the solution of single set of ODEs in pseudo-time. Neither requires matrix inversions except for the frequently diagonal observation error covariance. We analyze the behavior of the ODEs involved in these formulations. We demonstrate that they stiffen for large magnitudes of the ratio of background to observational error covariance, and that using the integration scheme proposed in both BGR09 and BR10 can lead to failure. An integration scheme that is both stable and is not computationally expensive is proposed. We develop transform-based alternatives for these Bucy-type approaches so that the integrations are computed in ensemble space where the variables are weights (of dimension equal to the ensemble size) rather than model variables. Finally, the performance of our ensemble transform Kalman-Bucy implementations is evaluated using three models: the 3-variable Lorenz 1963 model, the 40-variable Lorenz 1996 model, and a medium complexity atmospheric general circulation model (AGCM) known as SPEEDY. The results from all three models are encouraging and warrant further exploration of these assimilation techniques.