TIDAL FLOW FORECASTING USING REDUCED RANK SQUARE-ROOT FILTERS

The Kalman filter algorithm can be used for many data assimilation pro blems. For large systems, that arise from discretizing partial differe ntial equations, the standard algorithm has huge computational and sto rage requirements. This makes direct use infeasible for many applicati ons. In addition numerical difficulties may arise if due to finite pre cision computations or approximations of the error covariance the requ irement that the error covariance should be positive semi-definite is violated. In this paper an approximation to the Kalman filter algorith m is suggested that solves these problems for many applications. The a lgorithm is based on a reduced rank approximation of the error covaria nce using a square root factorization. The use of the factorization en sures that the error covariance matrix remains positive semi-definite at all times, while the smaller rank reduces the number of computation s and storage requirements. The number of computations and storage req uired depend on the problem at hand, but will typically be orders of m agnitude smaller than for the full Kalman filter without significant l oss of accuracy. The algorithm is applied to a model based on a linear ized version of the two-dimensional shallow water equations for the pr ediction of tides and storm surges. For non-linear models the reduced rank square root algorithm can be extended in a similar way as the ext ended Kalman filter approach. Moreover, by introducing a finite differ ence approximation to the Reduced Rank Square Root algorithm it is pos sible to prevent the use of a tangent linear model for the propagation of the error covariance, which poses a large implementational effort in case an extended kalman filter is used.