Helmert transformation

About: Helmert transformation is a research topic. Over the lifetime, 174 publications have been published within this topic receiving 1435 citations. The topic is also known as: seven-parameter transformation.

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 Squar...

The non-linear 2D symmetric helmert transformation : An exact non-linear least-squares solution

Abstract: In this paper a particular class of non-linear least-squares problems for which it is possible to take advantage of the special structure of the non-linear model, is discussed. The non-linear models are of the ruled-type (Teunisson, 1985a). The proposed solution strategy is applied to the2D non-linear Symmetric Helmert transformation which is defined in the paper. An exact non-linear least-squares solution, using a rotational invariant covariance structure is given.

Downward continuation of Helmert's gravity

Abstract: . The aim of this contribution is to show that mean Helmert's gravity anomalies obtained at the earth surface on a grid of a `reasonable' step can be transferred to corresponding mean Helmert's anomalies on the geoid. To demonstrate this, we take the \(\) by \(\) mean Helmert's anomalies from a very rugged region, the south-western corner of Canada which contains the two main chains of the Canadian Rocky Mountains, and formulate the problem of downward continuation of Helmert's anomalies for this region. This can be done exactly because Helmert's disturbing potential is harmonic everywhere outside the geoid, therefore even within the topography. Then we solve the problem numerically by transforming the Poisson integral to a system of 53,856 linear algebraic equations. Since the matrix of this system is well conditioned, there is no theoretical obstacle to the solution. The correctness of the solution is then checked by back substitution and by evaluating the contribution of the downward continuation term to Helmert's co-geoid. This contribution comes out positive for all the points. We thus claim that the determination of the downward continuation of mean Helmert's gravity anomalies on a grid of a `reasonable' step is a well posed problem with a unique solution and can be done routinely to any accuracy desired in the geoid computaion.