Evaluating Methods to Account for System Errors in Ensemble Data Assimilation

Jeffrey S. Whitaker NOAA/Earth System Research Laboratory/Physical Sciences Division, Boulder, Colorado

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Thomas M. Hamill NOAA/Earth System Research Laboratory/Physical Sciences Division, Boulder, Colorado

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Abstract

Inflation of ensemble perturbations is employed in ensemble Kalman filters to account for unrepresented error sources. The authors propose a multiplicative inflation algorithm that inflates the posterior ensemble in proportion to the amount that observations reduce the ensemble spread, resulting in more inflation in regions of dense observations. This is justified since the posterior ensemble variance is more affected by sampling errors in these regions. The algorithm is similar to the “relaxation to prior” algorithm proposed by Zhang et al., but it relaxes the posterior ensemble spread back to the prior instead of the posterior ensemble perturbations.

The new inflation algorithm is compared to the method of Zhang et al. and simple constant covariance inflation using a two-level primitive equation model in an environment that includes model error. The new method performs somewhat better, although the method of Zhang et al. produces more balanced analyses whose ensemble spread grows faster. Combining the new multiplicative inflation algorithm with additive inflation is found to be superior to either of the methods used separately.

Tests with large and small ensembles, with and without model error, suggest that multiplicative inflation is better suited to account for unrepresented observation-network-dependent assimilation errors such as sampling error, while model errors, which do not depend on the observing network, are better treated by additive inflation. A combination of additive and multiplicative inflation can provide a baseline for evaluating more sophisticated stochastic treatments of unrepresented background errors. This is demonstrated by comparing the performance of a stochastic kinetic energy backscatter scheme with additive inflation as a parameterization of model error.

Corresponding author address: Jeffrey S. Whitaker, NOAA/Earth System Research Laboratory/Physical Sciences Division, 325 Broadway R/PSD1, Boulder, CO 80305–3328. E-mail: jeffrey.s.whitaker@noaa.gov

Abstract

Inflation of ensemble perturbations is employed in ensemble Kalman filters to account for unrepresented error sources. The authors propose a multiplicative inflation algorithm that inflates the posterior ensemble in proportion to the amount that observations reduce the ensemble spread, resulting in more inflation in regions of dense observations. This is justified since the posterior ensemble variance is more affected by sampling errors in these regions. The algorithm is similar to the “relaxation to prior” algorithm proposed by Zhang et al., but it relaxes the posterior ensemble spread back to the prior instead of the posterior ensemble perturbations.

The new inflation algorithm is compared to the method of Zhang et al. and simple constant covariance inflation using a two-level primitive equation model in an environment that includes model error. The new method performs somewhat better, although the method of Zhang et al. produces more balanced analyses whose ensemble spread grows faster. Combining the new multiplicative inflation algorithm with additive inflation is found to be superior to either of the methods used separately.

Tests with large and small ensembles, with and without model error, suggest that multiplicative inflation is better suited to account for unrepresented observation-network-dependent assimilation errors such as sampling error, while model errors, which do not depend on the observing network, are better treated by additive inflation. A combination of additive and multiplicative inflation can provide a baseline for evaluating more sophisticated stochastic treatments of unrepresented background errors. This is demonstrated by comparing the performance of a stochastic kinetic energy backscatter scheme with additive inflation as a parameterization of model error.

Corresponding author address: Jeffrey S. Whitaker, NOAA/Earth System Research Laboratory/Physical Sciences Division, 325 Broadway R/PSD1, Boulder, CO 80305–3328. E-mail: jeffrey.s.whitaker@noaa.gov
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