Spatial decomposition of covariance functions in Generalized Kalman filter method for data assimilation

In this work, we consider the method of spatial decomposition of covariance functions in the Generalized Kalman method for data assimilation (GKFDA) proposed earlier. The standard expansion technique of covariance functions into eigenvectors and eigenvalues (Karhunen-Loeve decomposition) is used. With the help of this technique, the data assimilation problem into a hydrodynamic model is solved and errors of results are estimated. For the numerical experiments, the known Nucleus for European Modelling of the Ocean (NEMO) model and the Altimetry Validation and Interpolation Satellite Ocean data (AVISO) are used. All numerical experiments have been performed on HPC K-60 in Keldysh Institute of Applied Mathematics of Russian Academy of Sciences. This scheme of expansion is numerically implemented and the results of numerical experiments are analyzed. It is shown how many and which vectors in the covariance matrix decomposition make the most significant contribution to the components of the total result in data assimilation. Also, their values in the North Atlantic are calculated, the energy and the contribution of the eigenvalues of these vectors are estimated. Further directions of developments of this algorithm are indicated.