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Planar coordinate transformation and its parameter estimation in the complex number field

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Abstract

In this paper, the planar coordinate transformation model in the complex field is constructed. It is more concise than the traditional planar coordinate transformation model in the real field, because it combines the x and y coordinates as a whole. In order to carry out the complex parameter estimation, the paper introduces the complex least squares and the complex Gauss-Jacobi combinatorial algorithm. A simulative numerical case study and an actual one are given to investigate the correctness and efficiency of the presented approaches. The results show that the complex least squares for the complex transformation model obtains the identical estimation as the conventional least squares for the real transformation model, thus it is a reliable least squares for the complex parameter estimation. However, for the case of gross error, it becomes invalid. On the contrary, the complex Gauss-Jacobi combinatorial algorithm is capable of detecting the gross error based on its parameter estimations of numerous combinations. Consequently, it is verified as a good robust estimation by the numerical case study.

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Acknowledgements

The study is supported by National Natural Science Foundation of China (Grant No. 41104009), Open Research Fund Program of the Key Laboratory of Geospace Environment and Geodesy, Ministry of Education, China (Grant No. 11-01-04), and the Open Foundation of the Key Laboratory of Precise Engineering and Industry Surveying, National Administration of Surveying, Mapping and Geoinformation of China (Grant No. PF2011-4). The study is completed in Wuhan University the Key Laboratory of Geospace Environment and Geodesy, Ministry of Education, China. The author is grateful to the support and good working atmosphere provided by his research team in China Three Gorges University.

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  1. Key Laboratory of Geological Hazards on Three Gorges Reservoir Area, Ministry of Education, China Three Gorges University, Yichang, China

    Huaien Zeng

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  1. Huaien Zeng
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Correspondence to Huaien Zeng.

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Zeng, H. Planar coordinate transformation and its parameter estimation in the complex number field. Acta Geod Geophys 49, 79–94 (2014). https://doi.org/10.1007/s40328-014-0040-1

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  • DOI: https://doi.org/10.1007/s40328-014-0040-1

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