Toward a unified approach to the total least-squares adjustment

In this paper, we analyze the general errors-in-variables (EIV) model, allowing both the uncertain coefficient matrix and the dispersion matrix to be rank-deficient. We derive the weighted total least-squares (WTLS) solution in the general case and find that with the model consistency condition: (1) If the coefficient matrix is of full column rank, the parameter vector and the residual vector can be uniquely determined independently of the singularity of the dispersion matrix, which naturally extends the Neitzel/Schaffrin rank condition (NSC) in previous work. (2) In the rank-deficient case, the estimable functions and the residual vector can be uniquely determined. As a result, a unified approach for WTLS is provided by using generalized inverse matrices (g-inverses) as a principal tool. This method is unified because it fully considers the generality of the model setup, such as singularity of the dispersion matrix and multicollinearity of the coefficient matrix. It is flexible because it does not require to distinguish different cases before the adjustment. We analyze two examples, including the adjustment of the translation elimination model, where the centralized coordinates for the symmetric transformation are applied, and the unified adjustment, where the higher-dimensional transformation model is explicitly compatible with the lower-dimensional transformation problem.