In deformation analyses, it is important to find a stable reference frame and therefore the stability of the possible reference points must be controlled. There are several methods to test such stability. The paper’s objective is to examine one of such methods, namely the method based on application of R-estimation, for its sensitivity to gross errors. The method in question applies three robust estimators, however, it is not robust itself. The robustness of the method depends on the number of unstable points (the fewer unstable points there are, the more robust is the proposed method). Such property makes it important to know how the estimates applied and the strategy itself respond to a gross error. The empirical influence functions (EIF) can provide necessary information and help to understand the response of the strategy for a gross error. The paper presents examples of EIFs of the estimates, their application in the strategy and describes how important and useful is such knowledge in practice.

The method that is proposed in the present paper is a special case of squared M split estimation. It concerns a direct estimation of the shift between the parameters of the functional models of geodetic observations. The shift in question may result from, for example, deformation of a geodetic network or other non-random disturbances that may influence coordinates of the network points. The paper also presents the example where such shift is identified with a phase displacement of a wave. The shift is estimated on the basis of wave observations and without any knowledge where such displacement took place. The estimates of the shift that are proposed in the paper are named Shift- M split estimators.

The paper presents the method of robust estimation of variance coefficient. The concept of YRestimation presented in (6] is generalised in case of dependent observations. The basis of the method is usage of reinforcement matrix which guarantees the robustness of the estimate. The reinforcement matrix which is closely connected with the weight function of M-estimation, gives a possibility to perform robust adjustment. Thus such a method is also presented. At last, an example is shown too.

The paper presents an attempt to assess how random errors and systematic errors in gravity data affect the quality of the geoid model when it is computed using the FFf technique. Three groups of numerical tests were conducted with the use of gravity anomalies for Poland on 2' x 2' and 5' x 5' grid and with simulating random and systematic errors. In the first test, the effect of random errors on calculated geoid undulations was investigated, in the second one - the effect of systematic errors, and in the last one - the combined effect of both random and systematic errors. The effect of density of data set on the propagated error in geoid height was also examined. The results of numerical tests made possible to evaluate the effect of random errors as well as systematic errors on the accuracy of computed geoid undulation. They were also useful in evaluating the quality of the gravimetric quasigeoid model for Poland.

The problem of outlying observations is very well-known in the surveying data processing. Outliers might have several sources, different magnitudes, and shares within the whole observation set. It means that it is not possible to propose one universal method to deal with such observations. There are two general approaches in such a context: data cleaning or robust estimation. For example, the robust M-estimation has found many practical applications. However, there are other options, such as R-estimation or the absolute M _split estimation. The latter method was created to be less sensitive to outliers than the squared M _split estimation (the basic variant of Msplit estimation). From the theoretical point of view, the absolute M _split estimation cannot be classified as a robust method; however, it was proved that it could be used in such a context under certain conditions. The paper presents the primary comparison between that method and a conventional robust M-estimation. The results show that the absolute M _split estimation predominates over the classical methods, especially when the percentage of outliers is high. Thus, that method might be used to process LiDAR data, including mismeasured points. Processing synthetic data from terrestrial laser scanning or airborne laser scanning confirms that the absolute M _{split / estimation can deal with outliers sufficiently.}

M _split estimation is a novel method developed to process observation sets that include two (or more) observation aggregations. The main objective of the method is to estimate the location parameters of each aggregation without any preliminary assumption concerning the division of the observation set into respective subsets. Up to now, two different variants of M _split estimation have been derived. The first and basic variant is the squared M _split estimation, which can be derived from the assumption about the normal distribution of observations. The second variant is the absolute M _split estimation, which generally refers to the least absolute deviation method. The main objective of the paper is to compare both variants of M _split estimation by showing similarities and differences between the methods. The main dissimilarity stems from the different influence functions, making the absolute M _split estimation less sensitive to gross errors of moderate magnitude. The empirical analyses presented confirm that conclusion and show that the accuracy of the methods is similar, in general. The absolute M _split estimation is more accurate than the squared M _split estimation for less accurate observations. In contrast, the squared M _split estimation is more accurate when the number of observations in aggregations differs much. Concerning all advantages and disadvantages of M _split estimation variants, we recommend using the absolute M _split estimation in most geodetic applications.