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.
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.
In monitoring vertical displacements in elongated structures (e.g. bridges, dams) by means of precise geometric levelling a reference base usually consists of two subgroups located on both ends of a monitored structure. The bigger the separation of the subgroups, the greater is the magnitude of undetectable displacement of one subgroup with respect to the other. With a focus on a method of observation differences the question arises which of the two basic types of computation datum, i.e. the elastic and the fixed, both applicable in this method, is more suitable in such a specific base configuration. To support the analysis of this problem, general relationships between displacements computed in elastic datum and in fixed datum are provided. They are followed by auxiliary relationships derived on the basis of transformation formulae for different computational bases in elastic datum. Furthermore, indices of base separation are proposed which can be helpful in the design of monitoring networks. A test network with simulated mutual displacements of the base subgroups, is used to investigate behaviour of the network with the fixed and the elastic datum being applied. Also, practical guidelines are given concerning data processing procedures for such specific monitoring networks. For big separation of base subgroups a non-routine procedure is recommended, aimed at facilitating specialist interpretation of monitoring results.
The paper presents the results of investigating the effect of increase of observation correlations on detectability and identifiability of a single gross error, the outlier test sensitivity and also the response-based measures of internal reliability of networks. To reduce in a research a practically incomputable number of possible test options when considering all the non-diagonal elements of the correlation matrix as variables, its simplest representation was used being a matrix with all non-diagonal elements of equal values, termed uniform correlation. By raising the common correlation value incrementally, a sequence of matrix configurations could be obtained corresponding to the increasing level of observation correlations. For each of the measures characterizing the above mentioned features of network reliability the effect is presented in a diagram form as a function of the increasing level of observation correlations. The influence of observation correlations on sensitivity of the w -test for correlated observations (Förstner 1983,Teunissen 2006) is investigated in comparison with the original Baarda’s w -test designated for uncorrelated observations, to determine the character of expected sensitivity degradation of the latter when used for correlated observations. The correlation effects obtained for different reliability measures exhibit mutual consistency in a satisfactory extent. As a by-product of the analyses, a simple formula valid for any arbitrary correlation matrix is proposed for transforming the Baarda’s w -test statistics into the w -test statistics for correlated observations.