Geodetic deformation analysis can be interpreted as a model selection problem. The null model indicates that no deformation has occurred. It is opposed to a number of alternative models, which stipulate different deformation patterns. A common way to select the right model is the usage of a statistical hypothesis test. However, since we have to test a series of deformation patterns, this must be a multiple test. As an alternative solution for the test problem, we propose the p-value approach. Another approach arises from information theory. Here, the Akaike information criterion (AIC) or some alternative is used to select an appropriate model for a given set of observations. Both approaches are discussed and applied to two test scenarios: A synthetic levelling network and the Delft test data set. It is demonstrated that they work but behave differently, sometimes even producing different results. Hypothesis tests are well-established in geodesy, but may suffer from an unfavourable choice of the decision error rates. The multiple test also suffers from statistical dependencies between the test statistics, which are neglected. Both problems are overcome by applying information criterions like AIC.

Outlier detection is one of the most important tasks in the analysis of measured quantities to ensure reliable results. In recent years, a variety of multi-sensor platforms has become available, which allow autonomous and continuous acquisition of large quantities of heterogeneous observations. Because the probability that such data sets contain outliers increases with the quantity of measured values, powerful methods are required to identify contaminated observations. In geodesy, the mean shift model (MS) is one of the most commonly used approaches for outlier detection. In addition to the MS model, there is an alternative approach with the model of variance inflation (VI). In this investigation the VI approach is derived in detail, truly maximizing the likelihood functions and examined for outlier detection of one or multiple outliers. In general, the variance inflation approach is non-linear, even if the null model is linear. Thus, an analytical solution does usually not exist, except in the case of repeated measurements. The test statistic is derived from the likelihood ratio (LR) of the models. The VI approach is compared with the MS model in terms of statistical power, identifiability of actual outliers, and numerical effort. The main purpose of this paper is to examine the performance of both approaches in order to derive recommendations for the practical application of outlier detection.

To evaluate the benefit of a measurement procedure onto the estimated parameters, the dispersion of the parameters is usually used. To draw objective conclusions, unbiased or at least almost unbiased estimates are required. In geodesy, most of the functional relations are nonlinear but the statistical properties of the estimates are usually obtained by a linearised substitute-problem. Since the statistical properties of linear models cannot be passed to the nonlinear case, the estimates are biased. In this contribution, the bias of the parameters as well as the bias of the dispersion in nonlinear implicit models is investigated, using a second-order Taylor expansion. Nonlinear implicit models are general models and are used, for instance, in the framework of surface-fitting or coordinate transformation, which considers errors for the coordinates in source and target system. The bias is introduced as a further indicator to validate the benefit of an adapted measurement process using more precise measuring instruments. Since some parametrisations yield an ill-posed problem, also the case of a singular equation system is investigated. To demonstrate the second-order effect onto the estimates, a best-fitting plane is adjusted under varying configurations. Such a configuration is recommended in evaluating uncertainties of optical 3D measuring systems, e.g. in the framework of the VDI/VDE 2634 guideline. The estimated bias is used as an indicator whether a large number of poor observations provides better results than a small but precise sample.

In this contribution it is shown how an extended uncertainty budget of the observations according to the Guide to the Expression of Uncertainty in Measurement (GUM) can be considered in adjustment computations. The extended uncertainty budget results from the combination of Type A standard uncertainties determined with statistical methods and Type B standard uncertainties derived with nonstatistical methods. Two solutions are investigated, namely the adjustment in the classical Gauss-Markov model and the adjustment in the Gauss-Markov model using Monte Carlo simulations for the consideration of the uncertainties of the observations. Numerical examples are given to show that an appropriate interpretation of the dispersion measures for the unknowns is particularly important in order to avoid misinterpretation of the results. Furthermore, the effects of changing the weights of the observations on the adjustment results are shown. Finally, practical advice for the consideration of an extended uncertainty budget of the observations in adjustment computations is given.