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Erschienen in:
2020 | OriginalPaper | Buchkapitel
15. The Numerical Treatment of Covariance Stationary Processes in Least Squares Collocation
verfasst von : Wolf-Dieter Schuh, Jan Martin Brockmann
Erschienen in: Mathematische Geodäsie/Mathematical Geodesy
Verlag: Springer Berlin Heidelberg
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Abstract
Digital sensors provide long series of equispaced and often strongly correlated measurements. A rigorous treatment of this huge set of correlated measurements in a collocation approach is a big challenge. Standard procedures – applied in a thoughtless brute force approach – fail because these techniques are not suitable to handle such huge systems.
In this article two different strategies, denoted as covariance approach and filter approach, to handle such huge systems are contrasted. In the covariance approach various decorrelation strategies based on different Cholesky approaches to factorize the variance/covariance matrices are reviewed. The focus is on arbitrary distributed data sets with a finite number of data. But also extensions to sparse systems resulting from finite covariance functions and on exploiting the Toeplitz structure which results in the case of equispaced systems are elaborated.
Apart from that filter approaches are discussed to perform a prewhitening strategy for the data and rearrange the whole model to work with this filtered data in a rigorous way. Here, the special focus is on autoregressive processes to model the correlations. Finite, causal, non-recursive filters are constructed as prewhitening filters for the data as well as the model. This approach is extreme efficient, but can only deal with infinite equispaced data sets.
In real data scenarios, finite sequences and data gaps must be handled as well. For the covariance approach this is straightforward but it is a serious problem for the filter approach. Therefore a combination of these approaches is constructed to select the best properties from each. Covariance matrices of equispaced data sets designed by recursively defined covariance sequences are represented by AR processes as well as by Cholesky factorized matrices. It is shown, that it is possible to switch between both strategies to get data gaps and the warm up phase for the filter approach under control.