Linear Filtering of Sample Covariance for Ensemble Data Assimilation: Application of Optimality Criteria for the Estimation of Four-Dimensional Localization Functions

The estimation of background error covariance matrices is a critical component of JCSDA data assimilation systems. Most advanced estimation algorithms are based on ensemble perturbed forecasts. Because of computational limitations, those ensembles are limited in size, and filtering of the random sampling noise is necessary. This treatment is usually performed through Schur filtering, which consists of an element-by-element product of the sampling covariance with a specified correlation (localization) matrix The effect of the localization is to zero out many of the spurious long-distance correlations that are artificially created by the sampling noise. This suppression of spurious ensemble correlations has the double benefit to render the ensemble-based forecast error covariance sparse while increasing its rank. In practice, localization makes the implementation of ensemble based 4-dimensional data assimilation systems computationally tractable.

In JCSDA’s Gridpoint Statistical Interpolation (GSI) data assimilation system the localization matrix is separable, i.e. can be applied separately in the vertical and horizontal directions, but homogeneous and isotropic in each direction. The same localization matrix is furthermore applied to all 3-dimensional variables at all times. There is no time localization and the GSI 3D and 4DEnVar implementation relies on only two adjustable parameters: a horizontal and a vertical correlation length scale. Separability, independence from the variables and absence of time localization, have been the key assumptions to an efficient implementation of current 4DEnVar operational systems.

Ménétrier et al. (2015) proposed a new approach that seeks to minimize the difference between the localized covariance matrix and the asymptotic covariance matrix that would be obtained with an infinite size ensemble. This new paradigm allows an analytical development to get an optimality criterion for the Schur filtering and, finally, an explicit formulation of the localization function at a fractional computing cost of the analysis.

This talk describes our attempt to implement this new approach into a unified framework able to handle the full suite of JCSDA models (atmospheric, oceanic and sea ice). The framework is built on the OOPS infrastructure and, in its first incarnation, will estimate optimal 3-dimensional localization lengths from full ensemble background covariance matrices.