Data Assimilation with a Climatologically Augmented Local Ensemble Transform Kalman Filter

Data assimilation aims to optimally combine a best-guess forecast of a system state, known as the background state and typically made with a computer model, with observations of the true system state during the analysis procedure. Traditionally, data assimilation methods could be classified either as a “variational” or as an “ensemble” method. Although ensemble methods provide a computationally tractable method for estimating flow-dependent background error covariances, these estimations are limited by their severe rank deficiency. A consequence of this attribute of ensemble methods is that, for small ensemble sizes, the ensemble will not always span enough directions to fully utilize observational information. We present here a new method, which we call our hybrid method, that aims to remedy this rank deficiency by utilizing additional ensemble members when computing the analysis. These additional ensemble members, which we call “static” ensemble members, are constructed from a sum of the mean of the dynamic ensemble that is forecasted from analysis time to analysis time and an ensemble of climatologically-derived perturbations. The climatologically-based static ensemble members help add rank and covariance information to the ensemble-derived covariance estimate used when updating the ensemble.

In our method, the additional static perturbations are derived from the leading eigenvectors of the background error covariance matrix, to allow the analysis to account for errors likely to occur in directions potentially not well represented by the dynamic ensemble members. Specifically, we approximate the background error covariance matrix B as B = ZZ^T, and construct the perturbations from the columns of Z. The square root matrix Z can be found from the eigendecomposition of B as Z = VD^(1/2), where V is a matrix whose columns hold the orthonormal eigenvectors of B, and D is a diagonal matrix whose entries are the eigenvalues of B. At the end of the forecast phase of each analysis cycle, once the new climatological ensemble members have been constructed, assimilation is performed on the collection of both dynamic and static ensemble members. For m dynamic ensemble members and k static ensemble members, the analysis produces a k + m member analysis ensemble; we take the first m ensemble members from the analysis ensemble to form our dynamic analysis ensemble, which is then integrated to the next analysis time, where the cycle may be repeated.

We test our hybrid method on a simple, one-dimensional chaotic model, the Lorenz 240- variable model (Lorenz 2005), performing data assimilation using the Local Ensemble Transform Kalman Filter (LETKF) (Hunt et al. 2007). We perform a series of perfect model experiments, with observations generated by adding gaussian white noise to truth model states at observation locations. We compare our hybrid method to the original LETKF, at constant (dynamic) ensemble size, and find that the hybrid method is able to achieve similar analysis accuracies (as measured by the root-mean square error of the analysis ensemble mean) when compared to the LETKF, but with (approximately 30%) fewer ensemble members.

We find these results encouraging and suggestive that tests be performed on larger-dimensional systems. On such systems adding ensemble members, and hence producing additional forecasts, can be a great computational burden. Our method, which takes advantage of a larger ensemble only during the analysis, can help shed some of this cost. By considering a higher-dimensional space during the analysis than would otherwise be available from a pure ensemble method, we are able to better correct the state estimate, and to do so while forecasting with fewer ensemble members. Our method allows climatological information to readily be incorporated into existing ensemble assimilation systems, while also avoiding the cost of running two assimilation systems, as is required by more traditional hybrid data assimilation approaches.

References

Hunt, B. R., E. J. Kostelich, and I. Szunyogh, 2007: Efficient data assimilation for spatiotem- poral chaos: A local ensemble transform Kalman filter. Physica D: Nonlinear

Lorenz, E. N., 2005: Designing chaotic models. Journal of the Atmospheric Sciences, 62, 1574–1587.