Ocean Dynamics (2019) 69:1217–1237 1221 Now the matrix is as follows: A?1 = ?(Ne ? 1)I + (HXf T)T R?1(HXf T) (4) of size (Ne ? 1) × (Ne ? 1) is computed. Here, ? is the so- called forgetting factor, which is chosen as 0 ? ? ? 1 and inflates the ensemble variance to stabilise the filter process. I is the identity matrix and H is the observation operator which computes the model equivalent to the observations so that one can write y = Hxf + ? where y is the observation vector of size Ny , xf is a forecast state vector and ? is the observation error, which is assumed to be Gaussian with observation error covariance matrix R. The weight matrix W in Eq. 1 is now computed as the sum of two terms as follows: W = W? + W?. (5) Here, W? contains in each column the vector as follows: w? = TA(HXf T)T R?1(y ? Hx?f ) (6) which performs the transformation of the ensemble mean, while the ensemble perturbations are transformed by the following: W? = ?Ne ? 1TA1/2TT . (7) Here A1/2 = US1/2UT is the symmetric square root of A computed from the eigenvalue decomposition A = USUT . The degrees of freedom provided by the ensemble are too small to successfully assimilate the large number of satellite observations. Due to this, the ESKTF is applied here with a localised analysis as for the LSEIK filter (Nerger et al. 2006). Namely, the model state of each vertical column of the model grid is updated separately taking only observations into account that lie within a specified influence radius around the water column. Further, the observations are weighted according to their distance to reduce the influence of remote observations and to generate a smooth analysis field. For the weighting, the inverse observation error covariance matrix in Eq. 4 is multiplied element-by-element with a diagonal matrix constructed using the regulated localisation of Nerger et al. (2012a) with a correlation function given by the fifth-order polynomial of Gaspari and Cohn (1999). This function mimics a Gaussian function and varies between one at zero distance and zero at the distance of the influence radius. Since the model uses nested grids with different resolu- tions, one has to adapt the localisation. Here, the influence radius is chosen according to the location of the observa- tion, as is depicted in Fig. 2. Thus, an observation located in the coarse grid is only taken into account for model grid points within the radius rg, while an observation located in the fine grid is only taking into account within the radius rf. Accordingly, the analysis update of a water column on the coarse grid also takes into account observations on the Fig. 2 Localisation in nested model grids: the currently updated grid point in the coarse model grid is marked by the black dot. The blue circle marks the radius rg for which observations on the coarse grid include the analysis grid point. For observations on the fine grid, the corresponding shorter radius rf is marked by the green circle fine grid (vice versa for the update on the fine grid) if the grid point is sufficiently close to the fine grid. This ensures a smooth transition of the analysis field across the boundary of both grids. 3.3 Observations In the experiments, satellite observations of the sea surface temperature are assimilated. These are measured with the advanced very high resolution radiometer (AVHRR) aboard polar orbiting NOAA satellites and processed by the BSH. Composites over 12 h are used which are interpolated onto the two nested model grids. The composites use the satellite information over the 12-hour time window before the analysis step. Given that the radiometer provides only data for clear-sky conditions, the data coverage can vary significantly as shown in Fig. 3. This is particularly noticeable in the rather small fine grid region for the German coastal regions, where even 12-hour time windows with zero coverage can exist. For the validation of the assimilation results, a data set of in situ data is used. The data set includes data from the International Council for the Exploration of the Sea (ICES Dataset on Ocean Hydrography. The International Council for the Exploration of the Sea, Copenhagen. 2016) and the German Oceanographic Data Center (DOD, http://