We present a new type of the EnKF for data assimilation in spatial models that uses diagonal approximation of the state covariance in the wavelet space to achieve adaptive localization. The efficiency of the new method is demonstrated on an example.
 This work examines which generalized covariance function when used in the stochastic approach produces the flattest possible estimate of an unknown function that is consistent with the data. Such an estimate is the plainest possible continuous function, thus in a sense eliminating details that are irrelevant or unsupported by data. The answer is found from the solution of the following variational problem: Determine the function that reproduces the data, has the smallest gradient (in the square norm sense), and has a gradient that vanishes at large distances from the observations. The generalized covariance functions are shown to be the Green's functions for the free-space Laplace equation: the linear distance, in one dimension; the logarithmic distance in two dimensions; and the inverse distance in three dimensions. It is demonstrated that they are appropriate covariance functions for intrinsic random fields, a modification is proposed to facilitate numerical implementation, and a couple of examples are presented to illustrate the applicability of the methodology.  Background error covariances can be estimated from an ensemble of forecast differences. The finite size of the ensemble induces a sampling noise in the calculated statistics. It is shown formally that a wavelet diagonal approach amounts to locally averaging the correlations, and its ability to spatially filter this sampling noise is thus investigated experimentally. This is first studied in a simple analytical one dimensional framework. The capacity of a wavelet diagonal approach to model the scale variations over the domain is illustrated. Moreover, the sampling noise appears to be better filtered than when only using a Schur filter, in particular for small ensembles. The filtering properties are then illustrated for an ensemble of M´  Abstract Orthonormal wavelet analysis (OWA) is a special form of wavelet analysis, especially suitable for analyzing spatial structures, such as atmospheric fields. For this purpose, OWA is much more efficient and accurate than the nonorthogonal wavelet transform (WT), which was introduced to the meteorological community recently and which is more suitable for time series analysis. Whereas the continuous WT is strictly correct only for infinite domains, OWA is derived from periodizing and discretizing the infinite-domain case and so is correct for periodic boundary conditions. Unlike Fourier spectra, OWA is not shift invariant. Nor is it equivariant like the WT; that is, the OWA output does not shift as its input shifts. Two remedies are to combine all possible shifts, known as the overcomplete, nonorthogonal shift equivariant WT, or else to use a “best shift,” known as best shift wavelet analysis. Although shift invariant and orthonormal w.r.t. arbitrary inputs, the latter’s optimization generally depend... The use of orthogonal wavelets for the representation of background error covariances over a limited area is studied. Each wavelet function contains both information on position and information on scale: using a diagonal correlation matrix in wavelet space thus gives the possibility of representing the local variations of correlation scale. To this end, a generalized family of orthogonal Meyer wavelets that are not restricted to dyadic domains (i.e., powers of 2) is introduced. A three-bases approach is used, which allows one to take advantage of the respective properties of the spectral, wavelet, and gridpoint spaces. While the implied local anisotropies are relatively small, the local changes in the two-dimensional length scale are rather well represented. Statistical and balance features of forecast errors are generally incorporated in the background constraint of variational data assimilation. Forecast error covariances are here estimated with a spectral approach and from a set of forecast differences; autocovariances are calculated with a nonseparable scheme, and multiple linear regressions are used in the formulation of cross covariances. Such an approach was first developed for global models; it is here adapted to ALADIN, a bi-Fourier high-resolution limited-area model, and extended to a multivariate study of humidity forecast errors. Results for autocovariances confirm the importance of nonseparability, in terms of both vertical variability of horizontal correlations and dependence of vertical correlations with horizontal scale; high-resolution spatial correlations are obtained, which should enable a high-resolution analysis. Moreover nonnegligible relationships are found between forecast errors of humidity and those of mass and wind fields.