Nonlinear Geospatial Frame Transformations in the Presence of Noisy Data

Abstract

This paper presents an extended least squares framework for geospatial frame transformation problems with nonlinear models in the presence of noisy data. The goal is to set up an analytic scheme for obtaining transformed coordinates in the target frame with improved accuracy over the classic stepwise approach that is commonly used in practice. The proposed methodology is based on the joint spatial adjustment of all observed coordinates from the respective frames (initial and target) in conjunction with their stochastic model over all points of interest. This allows us to properly handle the cross-correlated noise in the initial coordinates between control and new points, and thus to overcome a crucial limitation of other operational approaches. In algebraic terms, the proposed approach relies on the weighted least squares adjustment of a stacked set of Gauss–Helmert models that can unequivocally describe any geospatial frame transformation over different groups of points. The analytic solution to this stacking problem is derived for the case of nonlinear parametric models, regardless of the structure of the error covariance matrices of the input data. The study focuses on computational aspects and the proper implementation of the solution algorithm via the Newton–Gauss iteration method. A numerical example is given to demonstrate the expected improvement in the statistical accuracy of the transformed coordinates under the proposed stacking approach.