# LETKF Scheme Description

The LETKF is one representative of a large family of ensemble Kalman filters which are collectively known as ensemble square-root (EnSRF) filters (

- Hunt, B. R. , E. Kalnay, E. J. Kostelich, E. Ott, D. J. Patil, T. Sauer, I. Szunyogh, J. A. Yorke, and A. V. Zimin, 2004: Four-dimensional ensemble Kalman filtering, Tellus A 56, 273–277.

- Hunt, B. R., E. Kostelich, I. Szunyogh, 2007: Efficient data assimilation for spatiotemporal chaos: a local ensemble transform Kalman filter. Physica D, 230, 112-126.

- Szunyogh I, Kostelich EJ, Gyarmati G, Kalnay E, Hunt BR, Ott E, Satterfield E, Yorke JA. 2008. A local ensemble transform Kalman filter data assimilation system for the NCEP global model. Tellus 60A: 113–130.

- Whitaker, J. S. and Hamill, T. M. 2012. Evaluating methods to account for system errors in ensemble data assimilation. Mon. Wea. Rev. 140, 3078_3089.

__Whitaker and Hamill, 2002__). Compared to other implementations of EnSRF filters, the LETKF offers some practical advantages for an operational NWP application. In the LETKF parallelization of the algorithm is natural and relatively straightforward to implement, computational scalability is theoretically close to 100%. Other noticeable advantages are the possibility to explicitly model observation covariances and to extend the filter to the temporal dimension (4D-LETKF,__Hunt et al., 2004__). The LETKF analysis ensemble mean is the linear combination of forecast ensemble states which best fits the observational dataset and is given by(1)

where

**X**and^{a}**X**, are the analysis and background ensemble mean respectively, and^{b}**X**the background ensemble perturbations (i.e., matrix whose columns store the differences between the background ensemble members and the background ensemble mean ). The analysis ensemble perturbations are updated by:^{b}(2)

where**P̃**

^{a}, differently from other versions of the EnSRF, the symmetric square root of

**P̃**

^{a}is used.

**P̃**

^{a}, the analysis error covariance in ensemble space, is given by

(3)

It is important to notice that if K is the ensemble size, the dimension of

**P̃**^{a}is K x K, usually much smaller than the dimension of both the model space and the number of available observations. This makes the LETKF algorithm computationally cheap. More details about the theoretical basis of LETKF are available in__Hunt et al. (2007) and Szunyogh et al. (2007)__. Multiplicative covariance inflation has been adopted as a simple and practical method to combat the insufficient ensemble spread arising from the small ensemble dimensionality and the failure to explicitly take into account model error. The inflation factor can be applied directly in (3) with the substitutionor by direct multiplication of to . The second solution has been adopted in the present work. The COMET implementation of the LETKF algorithm makes use of a 40 member ensemble based on the High-Resolution Regional Model (COSMO) which is updated every 3 hours with fresh observational data. The COSMO model is integrated on the Euro-Atlantic domain at 7 Km horizontal resolution on 49 vertical levels. In this way 40 analysis states

**x**are integrated forward making use of the full nonlinear atmospheric model to provide 40 forecast states_{a}^{i}(t)**x**at time t0+3h. This ensemble of forecast states provides us with the ensemble mean and covariances which are then updated according to Eqs. 1-3. The control variables are temperature, zonal and meridional wind components ,surface pressure and pseudo-relative humidity. Analysis of soil moisture is under testing in the framework of the EUMETSAT Fellowship program. In our implementation each grid point is influenced by all observations available in a local region with radius varying adaptively, in a way that the effective (weighted by their distance)number of observations in that radius is comparable to the number of degree of freedom of the system. Following_{b}^{i}(t)__Szunyiogh et al., 2008__, the weight of each observation on the analyzed grid point is decreased with distance r through multiplication of the**R**observation error matrix entries in Eqs. 1,3 with a smoothly decaying function of r. Localization is also imposed on the vertical coordinate.^{-1}**References:**- Hunt, B. R. , E. Kalnay, E. J. Kostelich, E. Ott, D. J. Patil, T. Sauer, I. Szunyogh, J. A. Yorke, and A. V. Zimin, 2004: Four-dimensional ensemble Kalman filtering, Tellus A 56, 273–277.

- Hunt, B. R., E. Kostelich, I. Szunyogh, 2007: Efficient data assimilation for spatiotemporal chaos: a local ensemble transform Kalman filter. Physica D, 230, 112-126.

- Szunyogh I, Kostelich EJ, Gyarmati G, Kalnay E, Hunt BR, Ott E, Satterfield E, Yorke JA. 2008. A local ensemble transform Kalman filter data assimilation system for the NCEP global model. Tellus 60A: 113–130.

- Whitaker, J. S. and Hamill, T. M. 2012. Evaluating methods to account for system errors in ensemble data assimilation. Mon. Wea. Rev. 140, 3078_3089.