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This paper concerns two types of Msplit estimation: squared Msplit estimation (SMS), which assumes normality of observation errors and absolute Msplit estimation (AMS), which applies {\text{L}_{1}} norm criterion. The main objective of the paper is to assess the accuracy of such estimators in vertical displacement analysis by applying Monte Carlo simulations. Another issue is to compare the accuracy of both estimators with the accuracy of the least squares estimation (LS). The paper shows that the accuracy of both Msplit estimates is like the accuracy of LS estimates. However, if some nonrandom errors occur, then accuracy of AMS estimates might be better than the accuracy of the rest of the estimates considered here. It stems from the fact that AMS estimates are robust against disturbances which have a small magnitude. It is also worth noting that the accuracy of both Msplit estimates might depend on the magnitude of the displacement.

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This paper presents an application of an Msplit estimation in the determination of terrain profiles from terrestrial laser scanning (TLS) data. We consider the squared Msplit estimation as well as the absolute Msplit estimation. Both variants have never been used to determine terrain profiles from TLS data (the absolute Msplit estimation has never been applied in any TLS data processing). The profiles are computed by applying polynomials of a different degree, determining which coefficients are estimated using the method in question. For comparison purposes, the profiles are also determined by applying a conventional least squares estimation. The analyses are based on simulated as well as real TLS data. The actual objects have been chosen to contain terrain details (or obstacles), which provide some measurements which are not referred to as terrain surface; here, they are regarded as outliers. The empirical tests prove that the proposed approach is efficient and can provide good terrain profiles even if there are outliers in an observation set. The best results are obtained when the absolute Msplit estimation is applied. One can suggest that this method can be used in a vertical displacement analysis in mining damages or ground disasters.

The main objective of the empirical influence function (EIF) is to describe how estimates behave when an observation set is affected by gross errors. Unlike the influence function, which represents the estimation method’s general properties, EIF can provide valuable information about applying different methods to a particular network. The chosen example allows us to compare different robust methods. The paper focuses on non-standard applications of EIF, for example, in assuming steering parameter of robust methods (usually related to the assumed interval for acceptable observation errors). The paper shows that commonly used values do not always work well, and EIFs might help choose appropriate values, guaranteeing the estimation process’s robustness. The most important new application of EIFs concerns the detection and assessment of a single gross error. The blinded experiments proved that such an approach is correct and can be an alternative to classic statistical tests for outlier detection.

Abstract. This paper presents practical aspect of the breakdown point theory in deformation analysis by applying R-estimators. The main aim of the paper is to determine impact of the probability of positive (or negative) gross errors and the number of such errors on the value of breakdown point of the estimates applied. Authors consider two types of networks: a levelling network and a horizontal one. Calculations are made for two cases, namely when observations are affected by gross errors in both measurement epochs or only in the second epoch. The main results are based on the Monte Carlo method, which is a very useful tool to solve such a geodetic problem. The simulations show that the breakdown point depends on the probability of positive gross errors but also on the number of epochs in which the gross errors occur. This is especially vivid in the case of levelling networks. Another interesting finding is that even if the number of gross errors exceeds the breakdown point, we can still get reasonable results; however, not always. Thus, the paper shows the probabilities that the method breaks down for several different cases. The paper includes some numerical tests, which provided practical information about the subjective breakdown points and their importance for R-estimates applied in deformation analysis.

Abstract:The determination of the accuracy of functions of measured or adjusted values may be a problem in geodetic computations. The general law of covariance propagation or in case of the uncorrelated observations the propagation of variance (or the Gaussian formula) are commonly used for that purpose. That approach is theoretically justifi ed for the linear functions. In case of the non-linear functions, the fi rst-order Taylor series expansion is usually used but that solution is affected by the expansion error. The aim of the study is to determine the applicability of the general variance propagation law in case of the non-linear functions used in basic geodetic computations. The paper presents errors which are a result of negligence of the higher-order expressions and it determines the range of such simplifi cation. The basis of that analysis is the comparison of the results obtained by the law of propagation of variance and the probabilistic approach, namely Monte Carlo simulations. Both methods are used to determine the accuracy of the following geodetic computations: the Cartesian coordinates of unknown point in the three-point resection problem, azimuths and distances of the Cartesian coordinates, height differences in the trigonometric and the geometric levelling. These simulations and the analysis of the results confi rm the possibility of applying the general law of variance propagation in basic geodetic computations even if the functions are non-linear. The only condition is the accuracy of observations, which cannot be too low. Generally, this is not a problem with using present geodetic instruments.