Variance Constraining Kalman Filtering
by
Lewis Mitchell
University of Sydney
Coauthors: Georg Gottwald, Sebastian Reich

Data assimilation aims to solve one of the fundamental problems of numerical weather prediction - estimating the optimal state of the atmosphere given a numerical model of the dynamics, and sparse, noisy observations of the system. The system is highly underdetermined due to the sparsity of available observations, so estimating the state is a non-trivial task.

We assume that the state of the system can be decomposed into ``observables'', for which observations are available, and ``pseudoobservables'', for which no direct observations are available. In atmospheric dynamics these pseudo-observables would correspond to fast gravity waves. Usually the number of pseudo-observables is orders of magnitude higher than for the observables. We consider the question how to incorporate a priori statistical information for the unobserved pseudo-observables in order to improve the analysis.

We develop a methodology for assimilating these pseudoobservations of the dynamics using an Ensemble Kalman filter (EnKF) - one of the state-of-the-art tools used in operational data assimilation.

We analyse this variance constraining Kalman filter (VCKF) for a simple linear toy model, showing analytically that the VCKF can produce improvements in forecast skill over existing EnKF methods. We also apply the VCKF to a higher-dimensional, non-linear dynamical system (Lorenz 1996) and show that incorporating information on the unobserved components of the system can again improve both the skill and stability of the data assimilation procedure. We show that the VCKF greatly improves upon the skill when observations are taken frequently, and when the observational noise is large.

Date received: January 4, 2010