Local Ensemble Transform Kalman Filter

Atmospheric dynamics is inherently local in space: the energy is generated, propagated and dissipated through local processes. Thus, errors in numerical model based estimations of the atmospheric state also propagate through local processes. In addition, we have found evidence, that at the scales a T62 (about 150 km) horizontal resolution and 28-level vertical resolution model can resolve, the atmosphere behaves locally like a low-dimensional chaotic system. This finding was first published in Patil et al. (2001) , a  study which was based on exploring the behavior of five independently generated bred vectors from the operational Global Forecast System (GFS) of the National Centers for environmental Prediction (NCEP)/National Weather Service. It was found, using the E-dimension statistic, that most of the uncertainty was associated with less than five directions,  which is the maximum number of independent state space directions a five-member ensemble could detect.

There were, however, two important limitations of the aforementioned study: (i) it was  based on a small five member ensemble, severely limiting the largest detectable dimension and (ii) bred vector ensembles are known to have a tendency to stay confined to low dimensional subspaces of the state space for extended periods. The first limitation, use of a small ensemble, was addressed by Oczkowski et al. (2005) , who studied much larger ensembles of bred vectors. This paper confirmed the major finding of Patil et al. (2001) that uncertainties were typically confined to a very few directions in the extra-tropics. The second limitation, the use of bred vectors, was addressed in Kuhl et al. (2006) , a paper in which the initial ensemble perturbations were generated with the LETKF. Since the LETKF tends to generate uniformly high dimensional uncertainty in the initial ensemble, a collapse of the uncertainties in the forecast ensembles can be attributed to the dynamics.

Our first attempt to exploit local low dimensionality of the atmospheric dynamics for efficient data assimilation was described in Ott et al. (2002). This paper was later revised and published both in a short letter format (Ott et al. 2004a) and in a full research article (Ott et al. 2004b) . The original scheme was named Local Ensemble Kalman Filter (LEKF). The most important unique feature of the LEKF was that it obtained the state estimate independently for each model grid point: it assimilated all observations that could potentially affect the state estimate at the given model grid point in one step. We note that in other flavors of the Ensemble Kalman Filter the observations are assimilated serially. In those schemes, the state estimate is updated successively, that is the state estimate at a given grid point can potentially until the assimilation of the observations has not been completed. The difference between our scheme and the others are primarily algorithmic: we expect to see more important differences in computational efficiency than in the accuracy of the state estimation.

The LEKF has been successfully tested with simulated observations on the National Centers for Environmental Prediction Global Forecast System (NCEP GFS) model (Szunyogh et al. 2005) . Preliminary results with real observations suggest that the LEKF scheme is competitive with the operational data assimilation system of NCEP. The LEKF has also been implemented on the Finite-Volume Global Circulation Model (FvGCM) of the NASA Goddard Space Flight Center. The performance of the LEKF was compared to that of the once operational PSAS data assimilation system of NASA by assimilating simulated radiosonde observations with both systems. We found that the LEKF was far superior to PSAS.

The Local Ensemble Transform Kalman Filter (LETKF), described in Hunt et al. (2007) is an enhanced version of the LEKF. The advantages of the LETKF is that (i) it is computationally more efficient (ii) it provides more flexibility in localization, which is needed for the assimilation of satellite radiances and (iii) it allows for a more straightforward implementation of the 4D extension of the scheme (Hunt et al. 2004) . The 4D extension allows for the assimilation of asynchronous observations using flow- (time-)dependent background error covariance information.

While our current implementation of the LETKF already provides accurate state estimates, we believe that including estimation of the model errors will further increase the accuracy of the scheme. We presented three different approaches to estimate a bulkmodel bias term within the LEKF-LETKF framework (Baek et al. 2006) . These schemes are based on the method of state augmentation and we are in the process of testing them with the operational models using real observations.