Noniterative Datum Transformation Revisited with Two-Dimensional Affine Model as a Case Study

In geospatial applications, the datum transformation has been necessarily employed to transform the geospatial outcomes from the data-collection system to the user-interested system. Its key is to compute the transformation parameters that describe the geometric relation between two datum systems. The ordinary least-squares based transformation parameter estimation needs the iterative computations unless the initial values of parameters are approximate enough, which is usually time-consuming. Particularly with the development of (near) real-time data collection techniques, such iterative datum transformation method cannot meet the real-time applications. In this paper, we study the noniterative method in terms of the multivariate least-squares theory with two-dimensional empirical affine transformation as a case study. We address the noniterative transformation for the partially and fully error-affected affine models, respectively. The study indicates that the noniterative solution exists when the variance matrix of coordinate errors is structured as Q0 circle times Q with Q the variance matrix of single point and Q0 the correlation matrix between points. The numerical examples show that the noniterative method can obtain the practically equivalent result with the ordinary method but improve the computation efficiency significantly. Therefore, the noniterative method is promising for the real-time datum transformation applications.