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Helmert transformation solutions combination and update with new measurements

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Abstract

This work is concerned with refining the Helmert solutions in two different cases, namely combining different solutions and updating old solutions with newly available measurement data. The parameters to be estimated in the solutions here not only include the transformation parameters, namely the rotation, the translation and the scale, but also the coordinates in the final frame of all relevant stations. The rotations are not limited to small angles, however, the rotation estimation errors are assumed with small angles. After developing in detail the functional model and the stochastic model of either of the two cases, the parameters are estimated directly using the least-squares method, with the full variance–covariance matrix of the parameter estimates readily provided as a by-product. These variances and covariances of/between the parameters fully capture the statistical information of the final solution, which are important components of the resulting frame network besides the parameters themselves. Simulations are conducted to check the method for both the combination and update cases. It is found that in both cases, the accuracies of the transformation parameters and coordinates of some stations are improved after the combination or update. There are also some stations whose coordinate estimation accuracies remain unchanged; and it is emphasized that including these stations is also necessary in the combination and update for completely describing final frame network.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (41774005, 41631072, 41474061).

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Authors and Affiliations

  1. Acoustic Science and Technology Laboratory, College of Underwater Acoustic Engineering, Harbin Engineering University, Harbin, 150001, China

    Shengquan Li

  2. Department of Navigation Engineering, Naval University of Engineering, Wuhan, 430033, China

    Bing Ji, Guobin Chang & Shaofeng Bian

  3. School of Environmental Science and Spatial Informatics, China University of Mining and Technology, Xuzhou, 221116, China

    Guobin Chang & Peng Lin

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  1. Shengquan Li
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  2. Bing Ji
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  3. Guobin Chang
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  4. Peng Lin
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  5. Shaofeng Bian
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Correspondence to Bing Ji.

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Li, S., Ji, B., Chang, G. et al. Helmert transformation solutions combination and update with new measurements. Acta Geod Geophys 54, 181–196 (2019). https://doi.org/10.1007/s40328-019-00249-5

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  • DOI: https://doi.org/10.1007/s40328-019-00249-5

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