The work presents the results of studies on dependence of effectiveness of chosen robust estimation methods from the internal reliability level of a geodetic network. The studies use computer-simulated observation systems, so it was possible to analyse many variants differing from each other in a planned way. Four methods of robust estimation have been chosen for the studies, differing substantially in the approach to weight modifications. For comparative reasons, the effectiveness studies have also been conducted for the very popular method in surveying practice, of gross error detection basing on LS estimation results, the so called iterative data snooping. The studies show that there is a relation between the level of network internal reliability and the effectiveness of robust estimation methods. In most cases, in which the observation contaminated by a gross error was characterized by a low index of internal reliability, the robust estimation led to results being essentially far from expectations.
The relationship between internal response-based reliability and conditionality is investigated for Gauss-Markov (GM) models with uncorrelated observations. The models with design matrices of full rank and of incomplete rank are taken into consideration. The formulas based on the Singular Value Decomposition (SVD) of the design matrix are derived which clearly indicate that the investigated concepts are independent of each other. The methods are presented of constructing for a given design matrix the matrices equivalent with respect to internal response-based reliability as well as the matrices equivalent with respect to conditionality. To analyze conditionality of GM models, in general being inconsistent systems, a substitute for condition number commonly used in numerical linear algebra is developed, called a pseudo-condition^number. Also on the basis of the SVD a formula for external reliability is proposed, being the 2-norm of a vector of parameter distortions induced by minimal detectable error in a particular observation. For systems with equal nonzero singular values of the design matrix, the formula can be expressed in terms of the index of internal response-based reliability and the pseudo-condition^number. With these measures appearing in explicit form, the formula shows, although only for the above specific systems, the character of the impact of internal response-based reliability and conditionality of the model upon its external reliability. Proofs for complementary properties concerning the pseudo-condition^number and the 2-norm of parameter distortions in systems with minimal constraints are given in the Appendices. Numerical examples are provided to illustrate the theory.
From the theory of reliability it follows that the greater the observational redundancy in a network, the higher is its level of internal reliability. However, taking into account physical nature of the measurement process one may notice that the planned additional observations may increase the number of potential gross errors in a network, not raising the internal reliability to the theoretically expected degree. Hence, it is necessary to set realistic limits for a sufficient number of observations in a network. An attempt to provide principles for finding such limits is undertaken in the present paper. An empirically obtained formula (Adamczewski 2003) called there the law of gross errors, determining the chances that a certain number of gross errors may occur in a network, was taken as a starting point in the analysis. With the aid of an auxiliary formula derived on the basis of the Gaussian law, the Adamczewski formula was modified to become an explicit function of the number of observations in a network. This made it possible to construct tools necessary for the analysis and finally, to formulate the guidelines for determining the upper-bounds for internal reliability indices. Since the Adamczewski formula was obtained for classical networks, the guidelines should be considered as an introductory proposal requiring verification with reference to modern measuring techniques.