A generalization of the analytical least-squares solution to the 3D symmetric Helmert coordinate transformation problem with an approximate error analysis

The symmetric Helmert transformation model is widely used in geospatial science and engineering. Using an analytical least-squares solution to the problem, a simple and approximate error analysis is developed. This error analysis follows the Pope procedure solving nonlinear problems, but no iteration is needed here. It is simple because it is not based on the direct and cumbersome error analysis of every single process involved in the analytical solution. It is approximate because it is valid only in the first-order approximation sense, or in other words, the error analysis is performed approximately on the tangent hyperplane at the estimates instead of the original nonlinear manifold of the observables. Though simple and approximate, this error analysis's consistency is not sacrificed as can be validated by Monte Carlo experiments. So the practically important variance-covariance matrix, as a consistent accuracy measure of the parameter estimate, is provided by the developed error analysis. Further, the developed theory can be easily generalized to other cases with more general assumptions about the measurement errors.