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On the consistency of the local ensemble square root Kalman filter perturbation update Cover

On the consistency of the local ensemble square root Kalman filter perturbation update

Tellus
Volume 71 (2019): Issue 1
By:  and    
Open Access
|Jan 2019
Figures & tables

Figures & Tables

Fig. 1.

Sequence of steps of a deterministic EnKF with covariance localisation, where the updated perturbations are obtained using the new scheme. Note that B and Pa need not be fully computed.

Fig. 2.

Density plots of the covariance matrices discussed in the text, except for P• and Pρ•. The raw sample covariance matrices are on the left, while the regularised (by localisation) sample covariance matrix are on the right. The true covariance matrix (B) cannot be visually discriminated from Pρ⋆ (bottom-right corner).

Fig. 3.

Plot of the Ne=8 perturbation sets: Xe,X̂,X⋆ and X•, with respect to the grid-point index.

Table 1.

Averaged Frobenius norm that measures the discrepancy between the target covariance matrix B and several raw (first row) or regularised (second row) sample error covariance matrices.

NormPeP̂P⋆P•||*−B||F19450331335||*ρ−B||F49490.050.06

[i] For the sake of comparison note that, on average, ||B||F=87.

Fig. 4.

Comparison of the LETKF, the LEnSRF and the LEnSRF with the new update scheme, applied to the L96 model (left column) and to the KS model (right column). The RMSE, optimal localisation and optimal inflation are plotted as functions of the ensemble size Ne.

Fig. 5.

Time-averaged RMSE as a function of the multiplicative inflation, the localisation length being tuned so as to minimise the RMSE. The L96 results are displayed on the left panels while the KS results are shown on the right panels, for Ne=4,8,16. An absent marker means that at least one of the 10 sample runs has diverged from the truth.

Fig. 6.

Comparison of the LETKF, the LEnSRF and the LEnSRF with the new update scheme, applied to the L96 model, for a fixed ensemble size Ne=8 and a fixed observation time step Δt=0.05. The RMSE (left panel) and the optimal inflation (right panel) are plotted as functions of the observation density Ny/Nx.

Fig. 7.

Comparison of the LETKF, the LEnSRF and the LEnSRF with the new update scheme, applied to the L96 model, for a fixed ensemble size Ne=8 and a fully observed model. The RMSE (left panel) and the optimal inflation (right panel) are plotted as functions of the observation time step Δt.

Fig. 8.

Time-averaged RMSE for the L96 model as a function of the multiplicative inflation, the localisation length being tuned so as to minimise the RMSE in the two configurations where the observations are sparser (Ny/Nx=0.50, left panel) and where the observations are infrequent (Δt=0.20, right panel). Ne=8 in both configurations.

Fig. 9.

Average analysis RMSE as a function of the norm p parameter in the range [1,11], and for Ne=4,8 and 16, applying the new LEnSRF scheme to the L96 model.

DOI: https://doi.org/10.1080/16000870.2019.1613142 | Journal eISSN: 3035-9554
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Language: English
Page range: 1613142 - 1613142
Published on: Jan 1, 2019
Published by: Stockholm University Press
In partnership with: Paradigm Publishing Services
Keywords:
data assimilation,
ensemble Kalman filter,
ensemble square root Kalman filter,
covariance localisation,
perturbation ensemble update

© 2019 Marc Bocquet, Alban Farchi, published by Stockholm University Press
This work is licensed under the Creative Commons Attribution 4.0 License.

Volume 71 (2019): Issue 1
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