Errors in the Initial Conditions for Numerical Weather Prediction: A Study of Error Growth Patterns and Error Reduction with Ensemble Filtering

Abstract

In this dissertation, we study the errors of a numerical weather prediction due to the errors in initial conditions and we present efficient nonlinear ensemble filters for reducing these errors.

First, we investigate the error growth, that is, the growth in time of the distance <em>E</em> between two solutions of a global weather model with similar initial conditions. Typically <em>E</em> grows until it reaches a saturation value <em>E_s</em>. We find two distinct broad <em>log-linear regimes</em>, one for <em>E</em> below 2% of <em>E_s</em> and the other for <em>E</em> above. In each, <em>log(E/E_s)</em> grows as if satisfying a linear differential equation. When plotting <em>dlog(E)/dt</em> vs <em>log(E)</em>, the graph is convex. We argue this behavior is quite different from error growth in other simpler dynamical systems, which yield concave graphs.

Secondly, we present an efficient variation of the Local Ensemble Kalman Filter (Ott et al. 2002, 2004) and the results of perfect model tests with the Lorenz-96 model. This scheme is locally similar to performing the Ensemble Transform Kalman Filter (Bishop et al. 2001). We also include a ``four-dimensional" extension of the scheme to allow for asynchronous observations.

Finally, we present a modified ensemble Kalman filter that allows a non-Gaussian background error distribution. Using a distribution that decays more slowly than a Gaussian is an alternative to using a high amount of variance inflation. We demonstrate the effectiveness of this approach for the three-dimensional Lorenz-63 model and the 40-dimensional Lorenz-96 model in cases when the observations are infrequent, for which the non-Gaussian filter reduces the average analysis error by about 10% compared to the analogous Gaussian filter. The mathematical formulation of this non-Gaussian filter is designed to preserve the computational efficiency of the local filter described in the previous paragraph for high-dimensional systems.