Total Least Squares Methods in Geodetic Transformations
Summary
Total Least Squares (TLS) methods have become integral to modern geodetic transformations, offering a rigorous framework for aligning coordinate datasets while accounting for measurement errors in all variables. Unlike ordinary least squares, which assumes error-free reference coordinates, TLS treats both source and target coordinates symmetrically, minimising orthogonal residuals and thereby reducing bias in parameter estimates. Over the past decade, extensions of TLS have introduced weight matrices to reflect heterogeneous uncertainties, leading to Weighted Total Least Squares (WTLS) formulations that improve the reliability of scale, rotation and translation parameters. These developments have been underpinned by the Errors-In-Variables (EIV) model and its embedding in the Gauss-Helmert adjustment scheme, enabling closed-form solutions and efficient iterative approaches. Applications range from datum shifts in national survey frameworks to point-cloud registrations in airborne LiDAR and photogrammetric networks. By delivering more accurate and uncertainty-aware transformations, TLS methods support high-precision tasks such as tectonic deformation monitoring, sea-level rise assessment and infrastructure mapping, thereby advancing the global geospatial community’s ability to integrate heterogeneous observation types with confidence.
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Total Least Squares Methods in Geodetic Transformations publication trend
The graph below shows the total number of articles in total least squares methods in geodetic transformations across all publications each year (not limited to Nature Index journals).
Technical terms
Total Least Squares (TLS): A regression technique that minimises orthogonal distances by allowing errors in both dependent and independent variables.
Weighted Total Least Squares (WTLS): An extension of TLS incorporating weight matrices to reflect varying uncertainties across observations.
Errors-In-Variables (EIV) Model: A statistical framework modelling errors in all measured quantities, forming the basis for TLS formulations.
Gauss-Helmert Model: A general linearised adjustment model accommodating both functional relationships and measurement errors in geodetic networks.
Similarity Transformation: A seven-parameter model comprising rotation, translation and uniform scale to align different coordinate systems.
References
- Weighted total least squares formulated by standard least squares theory. Journal of Geodetic Science (2012).
- On The Errors-In-Variables Model With Singular Dispersion Matrices. Journal of Geodetic Science (2014).
- WTLS iterative algorithm of 3D similarity coordinate transformation based on Gibbs vectors. Earth, Planets and Space (2020).
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