Fast recursive reconstruction of large time varying multidimensional fields

We develop computationally fast and storage efficient implementations for the Kalman-Bucy filter (KBf) for data assimilation problems with large time varying multidimensional fields. We refer to them as the block KBf (bKBf) and the localized block KBf (lbKBf). For fields defined on a 2D lattice of linear dimension I, the bKBf reduces the computational complexity of the KBf by O(I). The lbKBf saves further on computations by a factor of I and decreases the storage requirements by O(I). We illustrate the IbKBf in assimilating satellite measurements in physical oceanography, presenting simulations for an equatorial beta plane.