Testing methods for adjustment models with constraints

The method of least squares is widely used to fit data to a mathematical model, and the model is generally formulated as a Gauss- Markov model. Constraints on the parameters lead to a constrained adjustment of the observations. The validity of such a model can be tested. However, testing a model uniformly and if necessary, simultaneously, for both biased observations and invalid constraints has not yet been described; nor has a quality description of such tests yet been described. Here a simple, general procedure to resolve this is developed. It includes the computation of minimal detectable biases for both observations and constraints. Methods to compute the test statistic in the presence of singular covariance matrices (inevitable in the proposed procedure) and rank deficient coefficient matrices have not been published before. In this paper, an overview is given of six such methods. Constraints can describe deterministic, unmeasured model elements. In geodetic deformation analysis, for example, the stability of points can be formulated as constraints, tested, and provided with a quantification of the minimal detectable deformation. The analysis does not require the points that constitute the geodetic datum to be stable. Two comprehensive examples illustrate the use in geodetic deformation analysis.