  A new type of ensemble Kalman filter is developed, which is based on
replacing the sample covariance in the analysis step by its diagonal in a
spectral basis. It is proved that this technique improves the aproximation of
the covariance when the covariance itself is diagonal in the spectral basis, as
is the case, e.g., for a second-order stationary random field and the Fourier
basis. The method is extended by wavelets to the case when the state variables
are random fields, which are not spatially homogeneous. Efficient
implementations by the fast Fourier transform (FFT) and discrete wavelet
transform (DWT) are presented for several types of observations, including
high-dimensional data given on a part of the domain, such as radar and
satellite images. Computational experiments confirm that the method performs
well on the Lorenz 96 problem and the shallow water equations with very small
ensembles and over multiple analysis cycles.
