Nonlinear least-squares solution for the multi-station stacking problem in realizing a terrestrial reference frame

Multi-station (or time series) stacking is a fundamental task in the realization of a terrestrial reference frame (TRF), which is, however, rank-deficient in nature due to the arbitrarily selected target frame. In practice, such a model is approximated as a linear form and the classical free network theory is applied. It is known that the one-step adjustment only works in cases where the nonlinearity (measured by curvature) is moderate, and the initial point is very good; for TRF, it requires the deformable networks with a small enough time span for the network shape to be nearly unaltered. However, these assumptions can be nullified for the cases such as large time span and the integration of some local survey results. To address these limitations, we propose to solve the geodetically meaningful and numerically exact least-squares (LS) solution for the multi-station stacking model. The contributions are summarized as follows:

• The original nonlinear LS objective for the multi-station stacking model is investigated and its characteristics are analyzed;
• The nonlinear Baarda’s S-transformation is formulated for such a problem, which transforms different LS solutions that share the same network configuration;
• Two ways to obtain the geodetically meaningful solution are proposed, i.e., the minimally-constrained solution and the nearest-solution, where the latter originates from the inner-constraint solution in the linear case.
• The iterative schemes to obtain the two types of solutions are derived, which are shown to be essentially the truncated Gauss-Newton method. In addition, some techniques such as finite differences are employed to enhance numerical stability.

The developed theory is verified by the real examples.