No 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. INTRODUCTIONFinding the coordinates of geodetic network points in different coordinate systems is a common application in surveying. Therefore, a set of points with known coordinates in both coordinate systems is used to obtain the mathematical relationship between two systems' origins and the axes. This problem has been formulated by F.R. Helmert for 2D coordinate transformation and is now known as Helmert Transformation. In case of 3D, it is called Similarity Transformation. Both 2D and 3D geometric transformations are very useful tools in both geodetic and photogrammetric practice.The mathematical relationship between the two different coordinate systems is given by the so-called transformation parameters. These parameters, in 2D planar coordinate transformation, are two shiftings along the two axes, scale and the rotation angle between the axes of two coordinate systems. In case of 3D transformation, the number of parameters is seven, namely, three shiftings, three rotation angles and one scale parameter. Once these parameters were calculated from the identical points of the two coordinate set, coordinates of any point in one coordinate system can be easily computed in other coordinate system. Calculation of the transformation parameters is based on the solution of a system of equations where coordinates of points in one system are regarded as observation vector and the transformation parameters are regarded as unknown vector.Let X-Y and ξ-η define two planar cartesian ...

No abstract

Abstract. Deformation analysis is one of the main research fields in geodesy. Deformation analysis process comprises measurement and analysis phases. Measurements can be collected using several techniques. The output of the evaluation of the measurements is mainly point positions. In the deformation analysis phase, the coordinate changes in the point positions are investigated. Several models or approaches can be employed for the analysis. One approach is based on a Helmert or similarity coordinate transformation where the displacements and the respective covariance matrix are transformed into a unique datum. Traditionally a Least Squares (LS) technique is used for the transformation procedure. Another approach that could be introduced as an alternative methodology is the Total Least Squares (TLS) that is considerably a new approach in geodetic applications. In this study, in order to determine point displacements, 3-D coordinate transformations based on the Helmert transformation model were carried out individually by the Least Squares (LS) and the Total Least Squares (TLS), respectively. The data used in this study was collected by GPS technique in a landslide area located nearby Istanbul. The results obtained from these two approaches have been compared.

No abstract