Spectral characteristics of Kalman filter systems for atmospheric data assimilation

Abstract

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. -from Authors