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A Structurally Localized Ensemble Kalman Filtering Approach

Boujemaa Ait-El-Fquih, Ibrahim Hoteit

TL;DR

This work introduces a new ensemble filtering approach, which is inherently localized, avoiding the need for any auxiliary localization technique, and is comparable to those of the EnKF and ETKF with already tuned localization, both in terms of computational burden and estimation accuracy.

Abstract

State-of-the-art ensemble Kalman filtering (EnKF) algorithms require incorporating localization techniques to cope with the rank deficiency and the inherited spurious correlations in their error covariance matrices. Localization techniques are mostly ad-hoc, based on some distances between the state and observation variables, requiring demanding manual tuning. This work introduces a new ensemble filtering approach, which is inherently localized, avoiding the need for any auxiliary localization technique. Instead of explicitly applying localization on ensembles, the idea is to first localize the continuous analysis probability density function (pdf) before ensemble sampling. The localization of the analysis pdf is performed through an approximation by a product of independent marginal pdfs corresponding to small partitions of the state vector, using the variational Bayesian optimization. These marginals are then sampled following stochastic EnKF and deterministic ensemble transform Kalman filtering (ETKF) procedures, using ensembles larger than the partitions' size. The resulting filters involve the same forecast steps as their standard EnKF and ETKF counterparts but different analysis steps, iteratively adjusting the EnKF and ETKF updates of each partition based on the ensemble means of the other partitions. Numerical experiments are conducted with the Lorenz-96 model under different scenarios to demonstrate the potential of the proposed filters. The new filters' performances are comparable to those of the EnKF and ETKF with already tuned localization, both in terms of computational burden and estimation accuracy.

A Structurally Localized Ensemble Kalman Filtering Approach

TL;DR

This work introduces a new ensemble filtering approach, which is inherently localized, avoiding the need for any auxiliary localization technique, and is comparable to those of the EnKF and ETKF with already tuned localization, both in terms of computational burden and estimation accuracy.

Abstract

State-of-the-art ensemble Kalman filtering (EnKF) algorithms require incorporating localization techniques to cope with the rank deficiency and the inherited spurious correlations in their error covariance matrices. Localization techniques are mostly ad-hoc, based on some distances between the state and observation variables, requiring demanding manual tuning. This work introduces a new ensemble filtering approach, which is inherently localized, avoiding the need for any auxiliary localization technique. Instead of explicitly applying localization on ensembles, the idea is to first localize the continuous analysis probability density function (pdf) before ensemble sampling. The localization of the analysis pdf is performed through an approximation by a product of independent marginal pdfs corresponding to small partitions of the state vector, using the variational Bayesian optimization. These marginals are then sampled following stochastic EnKF and deterministic ensemble transform Kalman filtering (ETKF) procedures, using ensembles larger than the partitions' size. The resulting filters involve the same forecast steps as their standard EnKF and ETKF counterparts but different analysis steps, iteratively adjusting the EnKF and ETKF updates of each partition based on the ensemble means of the other partitions. Numerical experiments are conducted with the Lorenz-96 model under different scenarios to demonstrate the potential of the proposed filters. The new filters' performances are comparable to those of the EnKF and ETKF with already tuned localization, both in terms of computational burden and estimation accuracy.
Paper Structure (20 sections, 35 equations, 12 figures, 3 tables, 2 algorithms)

This paper contains 20 sections, 35 equations, 12 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Classical Ensemble Kalman Filtering: SEnKF and ETKF
  3. Problem formulation
  4. Ensemble Gaussian filtering: SEnKF and ETKF
  5. The (stochastic) SEnKF analysis step
  6. The (deterministic) ETKF analysis step
  7. Localization of the theoretical analysis pdf
  8. Calculation of the localized analysis pdfs $\pi_n({\bf x}_n^k)$
  9. The generic localized filtering algorithm
  10. Practical implementation: The partitioned-SEnKF (pSEnKF) and -ETKF (pETKF)
  11. The pSEnKF analysis step
  12. The pETKF analysis step
  13. Discussion
  14. Numerical experiments
  15. Full observational scenario
  16. ...and 5 more sections

Figures (12)

  • Figure 1: Evolution of the RSMN of the state analysis estimate as function of the iterations' number after every ${\rm N/3}$ assimilation cycles, with ${\rm N}$ being their total number.
  • Figure 2: Tracking of the first $4$ state variables with pSEnKF and pETKF within the last $100$ assimilation cycles.
  • Figure 3: Time-evolution of the MSE of the analysis state estimate as provided by pSEnKF and pETKF within the last $100$ assimilation cycles.
  • Figure 4: Averaged ${\rm RMSEs}$ of the state analysis estimates as provided by SEnKF, ETKF, pSEnKF and pETKF applied, with different ensemble sizes, on observations of ${\rm SNR} = 15$ dB (Panel (a)) and $10$ dB (Panel (b)). Panel (c) reports the ratios of ${\rm RMSEs}$ in Panel (b) w.r.t. those in Panel (a).
  • Figure 5: As in Fig. \ref{['figure_4']}, but with observing odd-indexed state variables only.
  • ...and 7 more figures