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Ensemble Kalman filtering with residual nudging Cover

Ensemble Kalman filtering with residual nudging

Tellus
Volume 64 (2012): Issue 1
By:  and    
Open Access
|Dec 2012
Figures & tables

Figures & Tables

Table 1. Time mean RMSEs and spreads of the KF, and the minimum time mean RMSEs (over different β) of the KF-RN, in the AR1 model with different S a

S a = 1 2 4 8 KF  RMSE 0.6184 0.8260 1.0592 1.2997  Spread 0.7729 1.0413 1.3419 1.8241 KF-RN  Min RMSE 0.6183 0.8259 1.0592 1.2997  Achieved at β=2 β=2 β≥2 β≥2

[i] The KF and KF-RN have identical time mean spreads; therefore, only those of the KF are presented. In the bottom row, we also report the ranges of β in which the minimum time mean RMSEs of the KF-RN are achieved.

Fig. 1. 

Time mean RMSEs of the KF and the KF-RN as functions of the noise level coefficient in the AR1 model, with different S a .

Fig. 2. 

Left panels: Sample time series of the fraction coefficients of the KF-RN with β=0.1 (upper) and β=1 (lower), respectively. Right panels: The corresponding histograms of the fraction coefficient time series.

Fig. 3. 

Time mean RMSEs of the normal EAKF and the EAKF-RN, as functions of inflation factor and half-width, in the full and 1/8 observation scenarios.

Table 2. Time mean RMSEs (spreads) of the normal EAKF and the EAKF-RN in the 1/2 observation scenario, as functions of the covariance inflation factor and the half-width of covariance localisation

l c =0.1 l c =0.2 l c =0.3 l c =0.4 l c =0.5 EAKF  λ=1.00 1.0721 (0.7049) Div Div Div Div  λ=1.05 1.0091 (0.7457) Div Div Div Div  λ=1.10 0.9789 (0.7868) Div Div Div Div  λ=1.15 0.9662 (0.8209) Div Div Div Div  λ=1.20 0.9515 (0.8566) Div Div Div Div  λ=1.25 0.9623 (0.8929) Div Div Div Div EAKF-RN  λ=1.00 1.0325 (0.7002) 1.8256 (0.5697) 2.1099 (0.5127) 2.2734 (0.4736) 2.2964 (0.4579)  λ=1.05 1.0051 (0.7419) 1.4072 (0.6185) 1.9879 (0.5644) 2.1821 (0.5269) 2.2468 (0.5050)  λ=1.10 0.9598 (0.7842) 1.2313 (0.6553) 1.8517 (0.6030) 2.0342 (0.5699) 2.1742 (0.5470)  λ=1.15 0.9673 (0.8201) 1.2024 (0.6870) 1.6507 (0.6388) 1.9317 (0.6015) 2.0953 (0.5845)  λ=1.20 0.9474 (0.8565) 1.1788 (0.7183) 1.5776 (0.6680) 1.9059 (0.6336) 2.0806 (0.6098)  λ=1.25 0.9650 (0.8935) 1.1856 (0.7484) 1.5315 (0.6945) 1.7778 (0.6603) 2.0071 (0.6383)

Table 3. As in Table 2, except that it is in the 1/4 observation scenario

l c =0.1 l c =0.2 l c =0.3 l c =0.4 l c =0.5 EAKF  λ=1.00 2.0685 (1.5730) Div Div Div Div  λ=1.05 1.9908 (1.7849) Div Div Div Div  λ=1.10 2.0223 (2.0447) 2.3014 (1.5640) Div Div Div  λ=1.15 2.0819 (2.3592) 2.2174 (1.7254) 2.9502 (1.5820) Div Div  λ=1.20 2.1903 (2.6869) 2.1839 (1.9468) 2.7534 (1.7191) Div Div  λ=1.25 2.3586 (3.0392) 2.2596 (2.2340) 2.6413 (1.8780) Div Div EAKF-RN  λ=1.00 2.0840 (1.5689) 2.6099 (1.1984) 3.0267 (1.0110) 3.0453 (0.8703) 3.0469 (0.7899)  λ=1.05 2.0042 (1.7790) 2.3341 (1.3762) 2.8493 (1.1936) 3.0573 (1.0403) 3.1015 (0.9618)  λ=1.10 1.9860 (2.0339) 2.2976 (1.5332) 2.8154 (1.3484) 3.0527 (1.2112) 3.1251 (1.1028)  λ=1.15 2.0766 (2.3648) 2.2389 (1.7244) 2.7737 (1.4940) 3.1247 (1.3341) 3.2583 (1.2558)  λ=1.20 2.1886 (2.6948) 2.2312 (1.9710) 2.6566 (1.6824) 3.0992 (1.5048) 3.2340 (1.3674)  λ=1.25 2.3436 (3.0359) 2.2352 (2.2344) 2.6168 (1.8427) 3.0977 (1.6509) 3.2897 (1.5098)
Fig. 4. 

Time mean spreads of the normal EAKF and the EAKF-RN, as functions of inflation factor and half-width, in the full and 1/8 observation scenarios.

Fig. 5. 

Time mean RMSEs of the normal EAKF and the EAKF-RN as functions of the noise level coefficient in different observation scenarios, with λ=1.15 and l c =0.1.

Fig. 6. 

As in Fig. 5, but with λ=1.05 and l c =0.3 for both the filters. Note that in the 1/2 and 1/4 observation scenarios divergences of the normal EAKF are spotted; hence, no horizontal lines are indicated in the corresponding plots. The EAKF-RN also diverges in the 1/2 and 1/4 observation scenarios for β≥4.

Fig. 7. 

Upper left: sample time series of the RMSE of the normal EAKF in the 1/2 observation scenario; upper right: sample time series of the RMSE of the EAKF-RN (β=2) under the same experiment settings as the EAKF; lower left: corresponding fraction coefficient c k in the EAKF-RN (β=2); lower right: corresponding histogram of c k .

Fig. 8. 

Upper: the RMSE of the EAKF (solid line with asterisks) and EAKF-RN (β=2, dotted line with plus signs) between the time instant k=1 and k=25; middle: difference in the RMSE (= RMSE of the EAKF – RMSE of the EAKF-RN) between k=1 and k=16; lower: the fraction coefficient of the EAKF-RN (β=2) between k=1 and k=25.

Fig. 9. 

Time mean RMSEs of the EAKF and the EAKF-RN, as functions of the ensemble size in different observation scenarios.

Fig. 10. 

Time mean RMSEs of the normal EAKF, as functions of the assimilation step S a and the observation noise variance, in different observation scenarios.

Fig. 11. 

As in Fig. 10, but for the EAKF-RN with β=2.

Fig. 12. 

Time mean RMSEs of the EAKF, as functions of the (possibly) mis-specified driving force F and the observation noise variance γ, in different observation scenarios.

Fig. 13. 

As in Fig. 12, but for the EAKF-RN with β=2.

DOI: https://doi.org/10.3402/tellusa.v64i0.17130 | Journal eISSN: 3035-9554
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Language: English
Page range: 17130 - 17130
Submitted on: Jan 2, 2012
Accepted on: Sep 6, 2012
Published on: Dec 1, 2012
Published by: Stockholm University Press
In partnership with: Paradigm Publishing Services
Keywords:
data assimilation,
ensemble Kalman filter,
residual nudging

© 2012 Xiaodong Luo, Ibrahim Hoteit, published by Stockholm University Press
This work is licensed under the Creative Commons Attribution 4.0 License.

Volume 64 (2012): Issue 1
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