A singular square root filter algorithm for large scale data assimilation

For systems with a relatively small order the Kalman filter provides an optimal solution to many data assimilation problems. However, for large scale systems that arise from discretizing partial differential equations, the number of computations and the required storage for the Kalman filter become prohibitive. Also numerical difficulties may arise from propagation of effors, due to finite machine precision computations. Often they result in the violation of the requirement that the error covariance matrix should be positive semi-definite. In this paper we will discuss the Singular Square Root Algorithm (SQRT) to compute a Kalman filter gain for large scale systems, which solves in a reasonable manner the problems mentioned above.