Bierman-Thornton UD Filtering for Double-Differenced Carrier Phase Estimation Accounting for Full Mathematical Correlation

David S. Chiu, Kyle P.G. O'Keffe

Abstract:	The Kalman filter is the standard method for estimating a kinematic GPS carrier phase float solution. Square-root filtering using modified Cholesky factors, also commonly known as the Bierman-Thornton UD algorithm, is an alternative to the standard Kalman filter. This method factorizes the state variance-covariance matrix into a set of matrices. These new factored matrices are propagated through the filter instead of the state variance-covariance matrix itself. More importantly, implementation of a UD factorization filter provides greater numerical stability in situations prone to roundoff error, large a priori errors and asymmetry of the state variance-covariance matrix, which may all lead to filter divergence. Having UD factors also makes it simple to pass on UD factors of the ambiguity states to LAMBDA. Doing so increases the efficiency of integer ambiguity estimation because LAMBDA’s first step is to decorrelate ambiguity states into such factors anyway. Furthermore, in our implementation, since UD matrices are factored only once (until there is a change in the makeup of the ambiguity states), the decorrelation of ambiguity states procedure is also done once, as opposed to LAMBDA, where such a decorrelation is performed at each epoch. To get further computational efficiency, decorrelation of the measurements can be done. This is a simple process where the measurement noise can be factorized into UDU T matrices. Then, substitutions are made into the measurement noise, design matrix and measurements using these factors. Once the measurements are decorrelated, it is possible to update the states and its variance-covariance matrix sequentially. There are several advantages of processing the measurements serially and they are discussed further in the paper. Theoretically, the numerical solution resulting from the Bierman-Thornton algorithm and the standard Kalman filter should be the same. Two datasets were used to assess and confirm this theory. Operational counts, quantified as flops, also show that the Bierman-Thornton method is computationally more efficient than the traditional filter.
Published in:	Proceedings of the 2008 National Technical Meeting of The Institute of Navigation January 28 - 30, 2008 The Catamaran Resort Hotel San Diego, CA
Pages:	756 - 762
Cite this article:	Chiu, David S., O'Keffe, Kyle P.G., "Bierman-Thornton UD Filtering for Double-Differenced Carrier Phase Estimation Accounting for Full Mathematical Correlation," Proceedings of the 2008 National Technical Meeting of The Institute of Navigation, San Diego, CA, January 2008, pp. 756-762.
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