Data assimilation in large time-varying multidimensional fields

(I)n the physical sciences, e.g., meteorology and oceanography, combining m easurements with the dynamics of the underlying models is usually referred to as data assimilation, Data assimilation improves the reconstruction of t he image fields of interest. Assimilating data with algorithms like the Kal man-Bucy filter (KBf) is challenging due to computational cost which for tw o-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 s cans) 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 KB f, and to O(I-4) in the case of the local block KBf (lbKBf) and the local s calar KBf (lsKBf), We illustrate the application of the lbKBf to assimilate altimetry satellite data in a Pacific equatorial basin.