Abstract
The Kalman filter is derived directly from the least-squares estimator, and generalized to accommodate stochastic processes with time variable memory. To complete the link between least-squares estimation and Kalman filtering of first-order Markov processes, a recursive algorithm is presented for the computation of the off-diagonal elements of the a posteriori least-squares error covariance. As a result of the algebraic equivalence of the two estimators, both approaches can fully benefit from the advantages implied by their individual perspectives. In particular, it is shown how Kalman filter solutions can be integrated into the normal equation formalism that is used for intra- and inter-technique combination of space geodetic data.
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Acknowledgments
The author thanks colleague Halfdan P. Kierulf for clarifying and stimulating discussions, and project member and colleague Michael Dähnn for feedback which made the work more readable. The author also thanks colleagues and project members Geir Arne Hjelle, Ingrid Fausk and Ann-Silje Kirkvik for their support. The presented work was facilitated by the management at the Geodetic Institute, Norwegian Mapping Authority, as part of its reference frame contribution project. The management is represented by Head of Section Oddgeir Kristiansen, Project Leader Laila Løvhøiden, Director of Geodetic Institute Per Erik Opseth and Assistant Director/Head of Section Reidun Kittelsrud. The computations were performed with Python 3 (www.python.org), using packages NumPy (www.numpy.org) and Matplotlib (Hunter 2007) (www.matplotlib.org).
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An erratum to this article is available at http://dx.doi.org/10.1007/s00190-017-1007-0.
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Mysen, E. On the equivalence of Kalman filtering and least-squares estimation. J Geod 91, 41–52 (2017). https://doi.org/10.1007/s00190-016-0936-3
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DOI: https://doi.org/10.1007/s00190-016-0936-3