On the construction of a reduced rank square-root Kalman filter for efficient uncertainty propagation

Abstract

The Kalman filter is a sequential estimation procedure that combines a stochastic dynamical model with observations in order to update the model state and the associated uncertainty. In the situation where no measurements are available the filter works as an uncertainty propagator. The most computationally demanding part of the Kalman filter is to propagate the covariance through the dynamical system, which may be completely infeasible in high-dimensional models. The reduced rank square-root (RRSQRT) filter is a special formulation of the Kalman filter for large-scale applications. In this formulation, the covariance matrix of the model state is expressed in a limited number of modes M. In the classical implementation of the RRSQRT filter the computational costs of the truncation step grow very fast with the number of modes (> M 3). In this work, a new approach based on the Lanzcos algorithm is formulated. It provides a more cost-efficient scheme and includes a precision coefficient that can be tuned for specific applications depending on the trade-off between precision and computational load.