Abstract
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.
Highlights
Data assimilation consists of incorporating new data periodically into computations in progress, which is of interest in many fields, including weather forecasting (e.g., Kalnay, 2003; Lahoz et al, 2010)
The ensemble Kalman filter (EnKF) (Evensen, 2009) replaces the state covariance by the sample covariance computed from an ensemble of simulations, which represent the state probability distribution
It can be proved that the EnKF converges to the KF in the large ensemble limit (Kwiatkowski and Mandel, 2015; Le Gland et al, 2011; Mandel et al, 2011) in the linear and Gaussian case, but an acceptable approximation may require hundreds of ensemble members (Evensen, 2009), because of spurious long-distance correlations in the sample covariance due to its low rank
Summary
Data assimilation consists of incorporating new data periodically into computations in progress, which is of interest in many fields, including weather forecasting (e.g., Kalnay, 2003; Lahoz et al, 2010). Further developments in the history of background covariance modeling in variational algorithms include construction of non-separable formulation (Courtier et al, 1998; Fisher and Andersson, 2001; Pannekoucke, 2009), representation of balances between variables in order to obtain a more realistic multivariate formulation (Derber and Bouttier, 1999; Fisher, 2003; Weaver et al, 2005), representation of heterogeneity using a physically/spectrally localized formulation (nonseparable wavelet formulation (Deckmyn and Berre, 2005; Fisher and Andersson, 2001), separable formulation based on diffusion operator (Weaver and Courtier, 2001) or recursive filters (Purser et al, 2003), and a non-separable formulation based on hybridization of diffusion and wavelets (Pannekoucke, 2009) Formulations such as the diffusion operator or the recursive filter are related to the diagonal assumption here, they involve covariance models with a relatively small number of parameters and free of sampling noise but estimated from an ensemble directly (Pannekoucke and Massart, 2008; Michel, 2013; Pannekoucke et al, 2014).
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