Data assimilation with correlated observation errors: experiments with a 1-D shallow water model

Laura M. Stewart; Sarah L. Dance; Nancy K. Nichols

doi:10.3402/tellusa.v65i0.19546

Data assimilation with correlated observation errors: experiments with a 1-D shallow water model

Tellus

Volume 65 (2013): Issue 1

By: Laura M. Stewart, Sarah L. Dance and Nancy K. Nichols

Open Access

|Dec 2013

Figures & Tables

Middle row of a 1001×1001 Markov matrix, (9). Dash–dot line L _R=0.01m, full line L _R=0.05m, dashed line L _R=0.1m and dotted line L _R=0.2 m.

(a) Eigenspectrum of a 1001×1001 Markov error correlation matrix; (b) Eigenspectrum of a 1001×1001 SOAR error correlation matrix.

Table 1. Analysis errors in u field at t=0 for different approximations to a Markov error covariance matrix ( ${‖ {\bar{x}}_{R} ‖}_{2} = 3.20$ )

ApproximationE1:

{‖ {\bar{x}}_{R_{f}} - x^{t} ‖}_{2}

{‖ {\bar{x}}_{R_{f}} - {\bar{x}}_{R_{t}} ‖}_{2}

E2 (%)Truth0.2000Diagonal0.300.237.22×diagonal0.310.237.24×diagonal0.310.247.5Markov (L_R=0.2)0.210.061.9Markov (L_R=0.1)0.2000Markov (L_R=0.05)0.210.051.6Markov (L_R=0.01)0.270.185.6ED (K=10)0.280.195.9ED (K=20)0.280.195.9ED (K=50)0.250.154.7ED (K=100)0.230.103.1

Table 2. Analysis errors in φ field at t=0 for different approximations to a Markov error covariance matrix ( ${‖ {\bar{x}}_{R} ‖}_{2} = 62.64$ )

ApproximationE1:

{‖ {\bar{x}}_{R_{f}} - x^{t} ‖}_{2}

{‖ {\bar{x}}_{R_{f}} - {\bar{x}}_{R_{t}} ‖}_{2}

E2 (%)Truth2.3500Diagonal3.613.044.92×diagonal3.853.325.34×diagonal4.113.615.8Markov (L_R=0.2)2.410.540.9Markov (L_R=0.1)2.3500Markov (L_R=0.05)2.420.671.1Markov (L_R=0.01)3.062.273.6ED (K=10)3.973.255.2ED (K=20)3.803.034.8ED (K=50)3.332.393.8ED (K=100)2.771.562.5

Table 3. Analysis errors in u field at t=0 for different approximations to a SOAR error covariance matrix ( ${‖ {\bar{x}}_{R} ‖}_{2} = 3.19$ )

ApproximationE1:

{‖ {\bar{x}}_{R_{f}} - x^{t} ‖}_{2}

{‖ {\bar{x}}_{R_{f}} - {\bar{x}}_{R_{t}} ‖}_{2}

E2 (%)Truth0.1100Diagonal0.310.288.82×diagonal0.320.299.14×diagonal0.320.309.4Markov (L_R=0.2)0.130.072.2Markov (L_R=0.1)0.150.113.4Markov (L_R=0.05)0.180.154.7Markov (L_R=0.01)0.270.257.8ED (K=10)0.260.247.5ED (K=20)0.230.206.3ED (K=50)0.150.113.4ED (K=100)0.130.072.2

Table 4. Analysis errors in φ field at t=0 for different approximations to a SOAR error covariance matrix ( ${‖ {\bar{x}}_{R} ‖}_{2} = 62.54$ )

ApproximationE1:

{‖ {\bar{x}}_{R_{f}} - x^{t} ‖}_{2}

{‖ {\bar{x}}_{R_{f}} - {\bar{x}}_{R_{t}} ‖}_{2}

E2 (%)Truth0.5700Diagonal3.363.325.32×diagonal3.593.555.74×diagonal3.993.956.3Markov (L_R=0.2)0.810.631.0Markov (L_R=0.1)1.181.061.7Markov (L_R=0.05)1.691.602.6Markov (L_R=0.01)2.892.844.5ED (K=10)3.903.876.2ED (K=20)3.713.675.9ED (K=50)1.561.452.3ED (K=100)1.060.851.4

Analysis errors in (a) u field and (b) φ field at the start of the time window. The grey line is for a diagonal approximation and the black line is for a Markov approximation with L _R=0.2m.

Analysis errors in (a) u field and (b) φ field at the centre of the time window, t=50. The grey line is for a diagonal approximation and the black line is for a Markov approximation with L _R=0.2m. Note that the scale in panel (b) is different from Fig. 3b.

Plot of E2 against level of observation noise for (a) u field, (b) φ field. The solid line is for the diagonal approximation, the dashed line for the ED approximation with K=50 and the dotted line for the Markov approximation with L _R=0.05m.

Table 5. Analysis errors in u field at t=0 for different approximations to a Markov observation error covariance matrix, with correlated background errors

ApproximationE1:

{‖ {\bar{x}}_{R_{f}} - x^{t} ‖}_{2}

{‖ {\bar{x}}_{R_{f}} - {\bar{x}}_{R_{t}} ‖}_{2}

E2 (%)Truth0.2300Diagonal0.300.195.72×diagonal0.280.165.04×diagonal0.260.144.4Markov (L_R=0.2)0.260.113.4Markov (L_R=0.1)0.2300Markov (L_R=0.05)0.230.072.3ED (K=50)0.340.216.4ED (K=100)0.300.165.0

Table 6. Analysis errors in φ field at t=0 for different approximations to a Markov observation error covariance matrix, with correlated background errors

ApproximationE1:

{‖ {\bar{x}}_{R_{f}} - x^{t} ‖}_{2}

{‖ {\bar{x}}_{R_{f}} - {\bar{x}}_{R_{t}} ‖}_{2}

E2 (%)Truth1.5900Diagonal2.071.372.22×diagonal2.061.372.24×diagonal2.061.352.2Markov (L_R=0.2)1.740.721.1Markov (L_R=0.1)1.5900Markov (L_R=0.05)1.650.450.7ED (K=50)1.850.911.4ED (K=100)1.770.781.3

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DOI: https://doi.org/10.3402/tellusa.v65i0.19546 | Journal eISSN: 3035-9554

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Language: English

Page range: 19546 - 19546

Submitted on: Aug 15, 2012

Accepted on: Mar 22, 2013

Published on: Dec 1, 2013

Published by: Stockholm University Press

In partnership with: Paradigm Publishing Services

Keywords:

variational data assimilation,

correlated observation errors,

approximate covariance matrices,

Markov correlation structures,

eigendecompositions,

shallow water equations

© 2013 Laura M. Stewart, Sarah L. Dance, Nancy K. Nichols, published by Stockholm University Press
This work is licensed under the Creative Commons Attribution 4.0 License.

Volume 65 (2013): Issue 1

Data assimilation with correlated observation errors: experiments with a 1-D shallow water model

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