A Stochastic Covariance Shrinkage Approach in Ensemble Transform Kalman Filtering

Andrey A. Popov; Adrian Sandu; Elias D. Nino-Ruiz; Geir Evensen

doi:10.16993/tellusa.214

A Stochastic Covariance Shrinkage Approach in Ensemble Transform Kalman Filtering

Tellus

Volume 75 (2023): Issue 1

By: Andrey A. Popov , Adrian Sandu , Elias D. Nino-Ruiz and Geir Evensen

Open Access

|Apr 2023

Figures & Tables

Algorithm 1 An explicit implementation of the stochastic shrinkage ETKF.
input: Forecast ensemble X^f, target matrix P, synthetic ensemble size M
output: Analysis ensemble X^a
1	$A^{f} \leftarrow \frac{1}{\sqrt{N - 1}} X^{f} (I_{N} - N^{- 1} 1_{N} 1_{N}^{T})$	`▷ Compute the forecast anomalies`
2	$Z^{f} \leftarrow \frac{1}{\sqrt{N - 1}} ℋ (X^{f}) (I_{N} - N^{- 1} 1_{N} 1_{N}^{T})$	`▷ Compute the observed forecast anomalies`
3	$A^{f} ~ N (0, P)$	`▷ Compute synthetic ensemble of anomalies`
4	$A^{f} \leftarrow \frac{1}{\sqrt{M - 1}} A^{f} (I_{M} - M^{- 1} 1_{M} 1_{M}^{T})$	`▷ Set mean to zero and scale`
5	$ χ^{f} \leftarrow \bar{X^{f}} 1_{M} + \sqrt{M - 1} A^{f}$	`▷ Compute the synthetic ensemble`
6	$Z^{f} = \frac{1}{\sqrt{M - 1}} ℋ ( χ^{f}) (I_{M} - M^{- 1} 1_{M} 1_{M}^{T})$	`▷ Compute the synthetic observed forecast anomalies`
7	$γ \leftarrow γ_{RBLW}$	`▷ Compute the shrinkage factor according to (2.15)`
8	$A^{f} \leftarrow [\begin{matrix} \sqrt{1 - γ} A^{f} & \sqrt{γ} A^{f} \end{matrix}]$	`▷ Construct the forecast anomaly ensemble`
9	$Z^{f} \leftarrow [\begin{matrix} \sqrt{1 - γ} Z^{f} & \sqrt{γ} Z^{f} \end{matrix}]$	`▷ Construct the obs. forecast anomaly ensemble`
10	$S \leftarrow Z^{f} Z^{f, T} + R$	`▷ Compute the combined oservation covariance`
11	$T = {(I_{(N + M) \times (N + M)} - Z^{f,T} S^{- 1} Z^{f})}^{\frac{1}{2}}$	`▷ Compute the transform matrix`
12	$A^{a} \leftarrow \frac{1}{\sqrt{1 - γ}} {[A^{f} T]}_{:, 1 : N}$	`▷ Compute the analysis anomalies`
13	${\bar{x}}^{a} \leftarrow \bar{X^{f}} + A^{f} T T^{T} Z^{f, T} R^{- 1} (y_{i}^{o} - \bar{ ℋ (X^{f})})$	`▷ Compute the analysis mean`
14	$X^{a} \leftarrow {\bar{x}}^{a} 1_{N}^{T} + \sqrt{N - 1} A^{a}$	`▷ Compute the analysis ensemble`

Results for the L96 problem with dynamic ensembles sizes of N = 5 and N = 14, inflation factor α = 1.1, and different synthetic ensemble sizes M. We compute the KL divergence of the rank histogram (4.2) and the RMSE (4.3) for the methods. Error bars show two standard deviations.

Results for the L96 problem. The left panel presents the analysis RMSE for various values of the dynamic and synthetic ensemble sizes. The right panel presents the shrinkage factor γ (2.15) for a synthetic ensemble size of M = 100 over a number of assimilation steps, with error bars showing two standard deviations.

Analysis RMSE results for the QG model. The experiments use a synthetic ensemble size M = 100 and Gaussian samples. Results are compared against LETKF with the Gaspari-Cohn decorrelation function (GC).

Left panel: initial condition of the water height with blue represented lower than average and yellow representing higher than average, and observation locations (red points). Right panel: analysis RMSE for the localized shrinkage ETKF, and the localized ETKF with the Gaspari-Cohn decorrelation function, with the error bars representing two standard deviations.

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DOI: https://doi.org/10.16993/tellusa.214 | Journal eISSN: 3035-9554

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Language: English

Page range: 159 - 171

Submitted on: Jul 12, 2022

Accepted on: Feb 2, 2023

Published on: Apr 11, 2023

Published by: Stockholm University Press

In partnership with: Paradigm Publishing Services

Keywords:

ensemble Kalman filter,

covariance shrinkage,

data assimilation,

synthetic ensemble,

localization

© 2023 Andrey A. Popov, Adrian Sandu, Elias D. Nino-Ruiz, Geir Evensen, published by Stockholm University Press
This work is licensed under the Creative Commons Attribution 4.0 License.

Volume 75 (2023): Issue 1

A Stochastic Covariance Shrinkage Approach in Ensemble Transform Kalman Filtering

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