Abstract.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 approximation 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.

Abstract. The ensemble Kalman smoother (EnKS) is used as a linear least-squares solver in the Gauss-Newton method for the large nonlinear least-squares system in incremental 4DVAR. The ensemble approach is naturally parallel over the ensemble members and no tangent or adjoint operators are needed. Furthermore, adding a regularization term results in replacing the Gauss-Newton method, which may diverge, by the Levenberg-Marquardt method, which is known to be convergent. The regularization is implemented efficiently as an additional observation in the EnKS. The method is illustrated on the Lorenz 63 model and a two-level quasigeostrophic model.

Bayesian inverse problem on an infinite dimensional separable Hilbert space with the whole state observed is well posed when the prior state distribution is a Gaussian probability measure and the data error covariance is a cylindrical Gaussian measure whose covariance has positive lower bound. If the state distribution and the data distribution are equivalent Gaussian probability measures, then the Bayesian posterior measure is not well defined. If the state covariance and the data error covariance commute, then the Bayesian posterior measure is well defined for all data vectors if and only if the data error covariance has positive lower bound, and the set of data vectors for which the Bayesian posterior measure is not well defined is dense if the data error covariance does not have positive lower bound.

No abstract