Total Least Squares Solution of Coordinate Transformation

Abstract

Coordinate transformation is one of the most commonly used processes in geodesy and surveying. Coordinates of points in one coordinate system are to be obtained in another coordinate system. To this end, the transformation parameters between two individual coordinate systems are calculated from the identical points, coordinates of which are known in both systems. This is achieved by the least-squares (LS) estimation. LS estimation is the classical approach in adjustment computations. It consists of a functional model that depicts the functional relation between the unknowns and the observations, and a stochastic model that represents the relative accuracies between the observations. In some cases, such as coordinate transformation, errors occur both in the observation vector and the design matrix. In classical approach, this is usually ignored and this ignorance remains as an uncertainty in the solution results. One way to take these errors in design matrix into account is to use Total Least Squares (TLS) estimation, which is quite new not only in surveying but also in mathematical sciences. By using TLS, one can take both the observations and all or a part of the design matrix as stochastic components. Therefore, more realistic values for the unknown parameters can be estimated. In this study, TLS technique was used to estimate the transformation parameters between two coordinate systems. The results are compared to the classical LS solution. TLS is able to handle the uncertainty and the results are more realistic than the classical approach.