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Table 1. Details of experiments executed using the Lorenz '96 (L) and Kuramoto-Sivashinsky (K) models to investigate the assimilation performance of the ETKFR compared to the ETKF
1LETKF (R=R t )50.682.3% (29.53)––2LETKF (R=diagR t )50.732.5% (29.53)––3LETKFR (R0=R D )50.702.4% (29.53)0.029.1% (0.22)4LETKF (R=R t )302.317.8% (29.46)––5LETKF (R=diagR t )302.839.6% (29.46)––6LETKFR (R0=R D )302.438.3% (29.46)0.0418.2% (0.22)7LETKF (R=R t )50.662.2% (29.53)––8LETKFR (R0=R D )50.672.3% (29.53)0.028.7% (0.23)1KETKF (R=R t )404.0019.0% (20.98)––2KETKF (R=diagR t )404.4121.0% (20.98)––3KETKFR (R0=R D )404.1219.6% (20.98)0.0517.2% (0.29)4KETKF (R=R t )1005.6526.8% (21.05)––5KETKF (R=diagR t )1006.0028.5% (21.05)––6KETKFR (R0=R D )1005.9528.3% (21.05)0.1448.2% (0.29)7KETKF (R=R t )403.8918.6% (20.98)––8KETKFR (R0=R D )404.0719.4% (20.98)0.0929.0% (0.31)
[i] The experiments are run with R t =R C +R D and . For experiments 1L to 6L the correlation length scale parameter is b=3.6 and for 1K to 6K is b=3.8. For 7L and 8L α=−3×10−4 and β=3.6 and for 7K and 8K α=3×10−4 and β=3.7. Note that for different runs of these experiments, the results are unchanged to two decimal places.
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Table 2. Details of experiments executed using the Lorenz '96 (L) and Kuramoto-Sivashinsky (K) models to investigate the robustness of the ETKFR
9Lα=0, β=5.00.10.742.5% (29.53)0.029.1% (0.22)10Lα=3×10−4, β=3.30.10.682.3% (29.53)0.029.1% (0.22)11Lα=−1×10−3, β=3.60.10.682.3% (29.53)0.0313.0% (0.23)12Lα=−3×10−4, β=3.60.010.210.7% (29.53)0.000.0% (0.02)13Lα=−3×10−4, β=3.61.02.438.2% (29.53)0.2611.6% (2.25)14Lα=−3×10−4, β=3.60.1, 1.0, 1.02.628.9% (29.53)0.2611.6% (2.25)15Lα=−3×10−4, β=3.61.0, 0.1, 0.10.682.3% (29.53)0.028.7% (0.23)9Kα=0, β=3.50.14.9723.7% (20.98)0.0725.0% (0.28)10Kα=−3×10−4, β=4.00.14.0319.2% (20.98)0.1034.5% (0.29)11Kα=4×10−4, β=3.70.14.0319.2% (20.98)0.1031.3% (0.32)12Kα=3×10−4, β=3.70.010.954.5% (20.98)0.000.0% (0.03)13Kα=3×10−4, β=3.71.011.2753.7% (20.98)0.8527.6% (3.08)14Kα=3×10−4, β=3.70.1, 1.0, 1.011.2553.6% (20.98)0.8828.6% (3.08)15Kα=3×10−4, β=3.71.0, 0.1, 0.14.1819.9% (20.98)0.1032.3% (0.31)
Fig. 1
Rows of the true and estimated covariance matrices. (a) Experiment 3L. Rows of the true (solid) and estimated covariance matrices. Covariance calculated using the first 100 background and analysis innovations (dashed). Covariance calculated using the last 100 background and analysis innovations (dot–dashed). (b) Experiment 3K. Rows of the true (solid) and estimated covariance matrices. Covariance calculated using the first 250 background and analysis innovations (dashed). Covariance calculated using the last 250 background and analysis innovations (dot–dashed).
Fig. 2
Rows of the true (solid) and estimated (dashed) covariance matrices (covariance function plotted against observation point) every 100 assimilation steps from 300 to 1000 for Experiment 8K with a time-dependent R, where b varies from 3.7 to 4.0, frequent observations and initial forecast, diagonal and correlated error variances set to 0.1.