Title: Errors in ensemble-based error covariance estimates and Hybrid ensemble 4D-VAR

Authors: Craig H. Bishop (Naval Research Laboratory, Monterey, CA)
Elizabeth Satterfield (Naval Research Laboratory, Monterey, CA)
David D Kuhl (Naval Research Laboratory, Washington DC.)
Tom Rosmond (Science Application International Corp., Forks, WA)

Ensemble Kalman filters produce predictions of flow dependent error covariances. Is it possible to empirically measure the accuracy of such predictions? How should such inaccuracy be accounted for in data assimilation? To address these questions, we use the Lorenz 40-variable model to help motivate an even simpler univariate forecasting system in which (i) the truth is a random draw from a climatological distribution of true states (ii) the forecast is the truth plus a random error drawn from a distribution having the true error variance (iii) the true error variance is a random draw from an inverse-gamma distribution, and (iv) the ensemble variance is a random draw from a gamma distribution whose mean is a linear function of the true forecast error variance. Bayes' theorem is used to obtain the distribution of true error variances given an ensemble variance. From this distribution, all aspects of the accuracy of the error variance prediction can be measured. Indeed, the approach allows the accuracy to be measured in terms of "effective ensemble size". It is found that the minimum error variance estimate of the true error variance given an ensemble variance is a weighted linear combination of the climatological variance and the ensemble variance. Hence, the result provides theoretical support for the use of linear combinations of static and ensemble based covariance models in Hybrid 4D-VAR schemes. If the univariate model's assumptions are valid, optimal weights for the ensemble and climatological variances can be recovered from large archives of (innovation, ensemble variance) pairs from any ensemble forecasting system. We make the ansatz that optimal weights for variances will make good weights for the static and ensemble based covariances of Hybrid ensemble 4D-VAR. Low resolution tests using a Hybrid ensemble version of the Navy's NAVDAS-AR data assimilation scheme have supported our ansatz. Limitations of our approach and possible ways of improving upon it will also be discussed.

GMAO Head: Michele Rienecker Global Modeling and Assimilation Office NASA Goddard Space Flight Center

Curator: Nikki Privé Last Updated: March 1 2011