Spectral Diagonal Covariance in Ensemble Kalman Filter

Abstract

Several variants of Ensemble Kalman filter (EnKF) based on Fast Fourier transform (FFT) or Discrete Wavelet transform (DWT) have been proposed recently. The main idea of these methods is to estimate the true forecast covariance matrix using only diagonal elements of sample covariance in spectral space, which reduce the amount of computations required by the EnKF, as well as the ensemble size. This approach is inspired by the fact that diagonal form of covariance implies, that the underlying random field is stationary for infinite dimensional grid and close to stationary for the finite domain. We will examine the proper conditions, under which the true covariance is diagonal and will show relationship between stationarity of underlying random field and the form of true covariance in spectral space. The analytical form of error of spectral diagonal covariance approximation, as a function of ensemble size, will be also derived. Simulations have shown, that the theoretical values of expected errors are obtained even for the really small ensemble.