Spectral Characteristics of Kalman Filter Systems for Atmospheric Data Assimilation

Abstract

In recent years, there has been increasing interest in the application of Kalman filter systems to atmospheric data assimilation. One important aspect of any data assimilation system is its filtering properties. This is examined by spectral decomposition of a simple one-dimensional Kalman filter system. It is shown that the second-moment error statistics of constant-coefficient linear systems observed everywhere on a regular grid are reduced to scalar systems by Fourier transforms. Under these conditions, the complete space and time behavior of the forecast and analysis error covariances can be explicitly determined from the model and observation error covariances and the initial forecast error covariance. The resulting solutions can then be examined by elementary dynamic systems analysis. The multivariate, inviscid, dissipative, unstable mode and nonstochastic cases are analyzed. The stationary solutions and the rate of convergence toward them are found and certain unstable, periodic, and quasi-periodic solutions are discussed. It is shown that the perfect, inviscid system converges to the stationary solution very slowly, but that viscosity improves the convergence rate. Wind insertion also improves the convergence-rate, but the wind only case is unstable. A simple method of determining model error statistics from forecast error statistics is discussed.