@techreport{VERLAAN96A,

author = {M. Verlaan},

title = {A Time Varying Kalman Filter for WAQUA},

institution = {Delft University of Technology},

year = {1996},

address = {Delft},

number = {96-74},

type = {TWI Technical report series},

note = {ftp: //ftp.twi.tudelft.nl

/TWI/publlications/tech-reports/1996},

annote = {In this report a new time varying approximate Kalman filter is

described, which can be used for large systems that arise from discretizing partial differential equations (eg. WAQUA). The algorithm uses a square root approach to avoid numerical difficulties, such as the violation of the requirement that the error covariance matrix should be positive semi-definite. The computational and storage requirements are reduced by approximating the error covariance with a matrix of lower rank. The rank of this matrix, which represents a balance between the accuracy of the estimate and the computational effort, can be chosen by the user. Experiments indicate that a rank (number of modes) of about 5-50 will already give accurate results, in contrast to the traditional 'full' Kalman filter which has a full rank of about 10000-100000 for typical models in

WAQUA. This implies that the number of computations needed for the new Reduced Rank Square Root (RRSQRT) algorithm will be reduced by about 100 to 10000 times compared to the

full time varying Kalman filter. The actual number of computations depends on the size of

the model and the number of modes needed to reach the specified accuracy.

The Reduced Rank Square Root (RRSQRT) algorithm can in principle be used for many data assimilation problems. In this report it was applied to a model, that was based on the two dimensional linearized shallow water equations. Also some tests were performed for a model that was model based on the (non-linear) shallow water equations. All tests indicated that the algorithm works well. Based on these results it is possible to implement the algorithm for use with the WAQUA system. For this purpose several aspects of the implementation are studied in this report as wel. For most parts of the implementation no difficulties are expected. However some care should be taken with implementation of the interface between the algorithm and the WAQUA system since WAQUA contains many dependencies (between variables) that are not very clear. For the implementation these dependencies should be made explicit.}

}