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Almost unbiased weighted least squares location estimation

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Abstract

Location service demands precise positioning, which can be obtained by applying maximum likelihood or least squares (LS) directly to measurements such as ranges and/or pseudoranges or to the computed quantities of measurements such as squared ranges and their differences. Nonlinearity and the stochastic models of computed quantities of measurements are not yet taken into account in precise location estimation. We propose a bias-corrected weighted LS method for precise location service, which consists of two basic elements. One is to automatically correct both the bias due to model nonlinearity and the induced biases in squared ranges/pseudoranges, and the other is to sequentially estimate unknown location parameters by treating equality constraints as a condition adjustment. As a result, the location estimation is theoretically almost unbiased. The method is applied to ranges, squared ranges and the differences of squared ranges. We also work out bias-corrected precise location estimation from pseudoranges, squared pseudoranges and the differences of pseudoranges. A large scale of simulations is carried out with range- and pseudorange-based models, respectively. The simulation results have clearly indicated: (i) the proposed bias-corrected weighted LS method can indeed result in almost unbiased estimates of location and further improve precise location estimation, depending on model nonlinearity and the ratio of signal to noise; (ii) the bias-corrected weighted LS method with the Gauss—Newton algorithm is shown to perform best in terms of the smallest bias and mean squared errors (MSE), if directly applied to ranges with a well-conditioned configuration, since it avoids the extra nonlinearity of squared ranges/pseudoranges and the induced bias of these computed quantities; (iii) the bias-corrected weighted LS and the weighted LS methods are of almost the same best performance in terms of MSE roots for each type of measurement models of ranges/pseudoranges, squared ranges/pseudoranges and the differences of squared ranges/pseudoranges. Nevertheless, the measurement models with the differences of squared ranges perform less accurate in the vertical component than those with ranges and squared ranges in the example, due to a weaker geometrical constraint; and (iv) the ordinary LS method clearly results in the most biased estimates of location and is of the worst accuracy for all the six measurement models of ranges/pseudoranges, squared ranges/pseudoranges and the differences of squared ranges/pseudoranges, indicating the importance of using correct stochastic models of measurements for location estimation.

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Data Availability

All the computations are based on simulations with the Matlab function randn to generate random numbers of standard normal distributions.

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Acknowledgements

This paper is dedicated in memory of Prof. Dr.-Ing.habil. Dr.tech.h.c.mult. Dr.-Ing.e.h.mult. Erik W. Grafarend (1939.10.30-2020.12.08). We thank the reviewers for their constructive comments. This work is partially supported by Chinese Academy of Engineering, Hubei Division, Project No. HB2020B13, and China National Natural Science Foundation, Project Nos. 42174045 and 41874012.

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Authors and Affiliations

  1. Disaster Prevention Research Institute, Kyoto University, Uji, Kyoto, 611-0011, Japan

    Peiliang Xu

  2. GNSS Research Center, Wuhan University, Wuhan, 430071, People’s Republic of China

    Jingnan Liu

  3. School of Geomatics, Xi’an University of Science and Technology, Xi’an, People’s Republic of China

    Yun Shi

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  1. Peiliang Xu
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Contributions

PX conceived the ideas and methodology, wrote the codes of computation, designed the experiments, performed computational analysis and wrote the manuscript. JL and YS participated in experimental designs and discussion of results and reviewed the manuscript.

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Correspondence to Peiliang Xu.

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Xu, P., Liu, J. & Shi, Y. Almost unbiased weighted least squares location estimation. J Geod 97, 68 (2023). https://doi.org/10.1007/s00190-023-01742-0

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