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On symmetrical three-dimensional datum conversion

Abstract

A 3-D similarity transformation is frequently used to convert GPS-WGS84-based coordinates to those in a local datum using a set of control points with coordinate values in both systems. In this application, the Gauss-Markov (GM) model is often employed to represent the problem, and a least-squares approach is used to compute the parameters within the mathematical model. However, the Gauss–Markov model considers the source coordinates in the data matrix (A) as fixed or error-free; this is an imprecise assumption since these coordinates are also measured quantities and include random errors. The errors-in-variables (EIV) model assumes that all the variables in the mathematical model are contaminated by random errors. This model may be solved using the relatively new total least-squares (TLS) estimation technique, introduced in 1980 by Golub and Van Loan. In this paper, the similarity transformation problem is analyzed with respect to the EIV model, and a novel algorithm is described to obtain the transformation parameters. It is proved that even with the EIV model, a closed form Procrustes approach can be employed to obtain the rotation matrix and translation parameters. The transformation scale may be calculated by solving the proper quadratic equation. A numerical example and a practical case study are presented to test this new algorithm and compare the EIV and the GM models.

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Acknowledgments

The authors would like to thank Burkhard Schaffrin, José A. Ramos, Alfred Leick, and the anonymous reviewer for their suggestions and assistance. The authors were partially supported in this research by the National Geospatial-Intelligence Agency under contract No. HM1582–04-1-2026.

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Correspondence to Yaron A. Felus.

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Felus, Y.A., Burtch, R.C. On symmetrical three-dimensional datum conversion. GPS Solut 13, 65 (2009). https://doi.org/10.1007/s10291-008-0100-5

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Keywords

  • Datum conversion
  • Total least-squares adjustment
  • Procrustes algorithm