Efficient Local Error Parameterizations for Square Root or Ensemble Kalman Filters: Application to a Basin-Scale Ocean Turbulent Flow

Abstract

Abstract In large-sized atmospheric or oceanic applications of square root or ensemble Kalman filters, it is often necessary to introduce the prior assumption that long-range correlations are negligible and force them to zero using a local parameterization, supplementing the ensemble or reduced-rank representation of the covariance. One classic algorithm to perform this operation consists of taking the Schur product of the ensemble covariance matrix by a local support correlation matrix. However, with this parameterization, the square root of the forecast error covariance matrix is no more directly available, so that any observational update algorithm requiring this square root must include an additional step to compute local square roots from the Schur product. This computation generates an additional numerical cost or produces high-rank square roots, which may deprive the observational update from its original efficiency. In this paper, it is shown how efficient local square root parameterizations can be obtained, for use with a specific square root formulation (i.e., eigenbasis algorithm) of the observational update. Comparisons with the classic algorithm are provided, mainly in terms of consistency, accuracy, and computational complexity. As an application, the resulting parameterization is used to estimate maps of dynamic topography characterizing a basin-scale ocean turbulent flow. Even with this moderate-sized system (a 2200-km-wide square basin with 100-km-wide mesoscale eddies), it is observed that more than 1000 ensemble members are necessary to faithfully represent the global correlation patterns, and that a local parameterization is needed to produce correct covariances with moderate-sized ensembles. Comparisons with the exact solution show that the use of local square roots is able to improve the accuracy of the updated ensemble mean and the consistency of the updated ensemble variance. With the eigenbasis algorithm, optimal adaptive estimates of scaling factors for the forecast and observation error covariance matrix can also be obtained locally at negligible additional numerical cost. Finally, a comparison of the overall computational cost illustrates the decisive advantage that efficient local square root parameterizations may have to deal simultaneously with a larger number of observations and avoid data thinning as much as possible.