Data assimilation in large time-varying multidimensional fields

(I)n the physical sciences, e.g., meteorology and oceanography, combining measurements with the dynamics of the underlying models is usually referred to as data assimilation, Data assimilation improves the reconstruction of the image fields of interest. Assimilating data with algorithms like the Kalman-Bucy filter (KBf) is challenging due to computational cost which for two-dimensional (2-D) fields is of O(I-6) where I is the linear dimension of the domain. In this paper, we combine the block structure of the underlying dynamical models and the sparseness of the measurements (e.g., satellite scans) to develop four efficient implementations of the KBf that reduce its computational cost to O(I-5) in the case of the block KBf and the scalar KBf, and to O(I-4) in the case of the local block KBf (lbKBf) and the local scalar KBf (lsKBf), We illustrate the application of the lbKBf to assimilate altimetry satellite data in a Pacific equatorial basin.