Spectral diagonal ensemble Kalman filters

TL;DR

This paper introduces a spectral diagonal ensemble Kalman filter that improves covariance approximation by using spectral basis diagonalization, with extensions to wavelets for non-homogeneous fields, and demonstrates efficiency in high-dimensional data assimilation.

Contribution

It develops a novel spectral basis approach for ensemble Kalman filtering, enhancing covariance approximation and extending to wavelet bases for complex fields.

Findings

Improved covariance approximation when the true covariance is diagonal in spectral basis.

Efficient implementation using FFT and DWT for high-dimensional data.

Successful application to Lorenz 96 and shallow water equations with small ensembles.

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 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.