| 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 E between two solutions of a global weather model with similar initial conditions. Typically E grows until it reaches a saturation value Es. We find two distinct broad log-linear regimes, one for E below 2% of Es and the other for E above. In each, log(E/Es) grows as if satisfying a linear differential equation. When plotting dlog(E)/dt vs log(E), 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 [32, 33] and the results of perfect model tests with the Lorenz-96 model. This scheme is locally similar to performing the Ensemble Transform Kalman Filter [5]. 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. |
