The paper addresses the problem of solving overdetermined systems of linear equations by means of methods of robust estimations, which eliminate the effect of outliers on the estimation results. The process of estimating a vector of parameters was accomplished by means of circular in structure neural networks. Formulating the problem in the aspect of a method for estimating parameters requires formulating an energy function (objective function) whose form was modified by means of a determined weighting function. In the final part of the paper the effectiveness of the methods described was evaluated in terms of controlling and diagnosing a geodetic observation system. The article is merely an introduction to a broadly understood problem of geodetic uses of robust estimators.

An increased use of global navigation techniques for positioning, and in particular for height determination, led to a growing need for precise models of height reference surface, i.e. geoid or quasigeoid. Geoid or quasigeoid heights at a cm accuracy level, provided on growing number ofGPS/levelling sites, can not only be used for quality control of gravimetric geoid but they also can be integrated with gravity data for geoid/quasigeoid modelling. Such a model is of particular use for surveying practice. A method of quasigeoid modelling based on GPS/levelling data with support of geopotential model and gravity data was developed. The components of height anomaly are modelled with the deterministic part that consists of height anomaly based on EGM96 geopotential model and Molodensky's integral, as well as the polynomial representing trend, and from the stochastic part represented by the isotropic covariance function. Model parameters, i.e. polynomial coefficients and covariance function parameters are determined in a single process of robust estimation, resistant to the outlying measurements. The method was verified using almost a thousand height anomalies from the sites of the EUREF-POL, POLREF, EUVN'97 and WSSG (Military Satellite Geodetic Network) networks in Poland as well as geopotential model refined with gravity data in l' x l' grid. The estimated average mean square error of quasigeoid height is at the level of O.Ol m. The outlying measurements were efficiently detected.

In the paper we present robust estimation methods based on bounded innovation propagation filters and quantile regression, applied to measure Value at Risk. To illustrate advantage connected with the robust methods, we compare VaR forecasts of several group of instruments in the period of high uncertainty on the financial markets with the ones modelled using traditional quasi-likelihood estimation. For comparative purpose we use three groups of tests i.e. based on Bernoulli trial models, on decision making aspect, and on the expected shortfall.

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 paper presents a summary of research activities concerning theoretical geodesy performed in Poland in the period of 2011–2014. It contains the results of research on new methods of the parameter estimation, a study on robustness properties of the M-estimation, control network and deformation analysis, and geodetic time series analysis. The main achievements in the geodetic parameter estimation involve a new model of the M-estimation with probabilistic models of geodetic observations, a new Shift-M split estimation, which allows to estimate a vector of parameter differences and the Shift- M split (+) that is a generalisation of Shift- M split estimation if the design matrix A of a functional model has not a full column rank. The new algorithms of the coordinates conversion between the Cartesian and geodetic coordinates, both on the rotational and triaxial ellipsoid can be mentioned as a highlights of the research of the last four years. New parameter estimation models developed have been adopted and successfully applied to the control network and deformation analysis. New algorithms based on the wavelet, Fourier and Hilbert transforms were applied to find time-frequency characteristics of geodetic and geophysical time series as well as time-frequency relations between them. Statistical properties of these time series are also presented using different statistical tests as well as 2 nd , 3 rd and 4 th moments about the mean. The new forecasts methods are presented which enable prediction of the considered time series in different frequency bands.

Generally, gross errors exist in observations, and they affect the accuracy of results. We review methods to detect the gross errors by Robust estimation method based on L1-estimation theory and their validity in adjustment of geodetic networks with different condition. In order to detect the gross errors, we transform the weight of accidental model into equivalent one using not standardized residual but residual of observation, and apply this method to adjustment computation of triangulation network, traverse network, satellite geodetic network and so on. In triangulation network, we use a method of transforming into equivalent weight by residual and detect gross error in parameter adjustment without and with condition. The result from proposed method is compared with the one from using standardized residual as equivalent weight. In traverse network, we decide the weight by Helmert variance component estimation, and then detect gross errors and compare by the same way with triangulation network In satellite geodetic network in which observations are correlated, we detect gross errors transforming into equivalent correlation matrix by residual and variance inflation factor and the result is also compared with the result from using standardized residual. The results of detection are shown that it is more convenient and effective to detect gross errors by residual in geodetic network adjustment of various forms than detection by standardized residual.

The summary of research activities concerning general theory and methodology performed in Poland in the period of 2015–2018 is presented as a national report for the 27th IUGG (International Union of Geodesy and Geophysics) General Assembly. It contains the results of research on new or improved methods and variants of robust parameter estimation and their application, especially to control network analysis. Reliability analysis of the observation system and an integrated adjustment approach are also given. The identifiability (ID) index as a new measure for minimal detectable bias (MDB) in the observation system of a network, has been introduced. A new method of covariance function parameter estimation in the least squares collocation has been developed. The robustified version of the Shift-Msplit estimation, termed as Shift-M*split estimation, which enables estimation of parameter differences (robustly), without the need of prior estimation of the parameters, has been introduced. Results on the analysis of geodetic time series, particularly Earth orientation parameter time series, geocenter time series, permanent station coordinates and sea level variation time series are also provided in this review paper. The entire bibliography of related works is provided in the references.

In the paper issues related to the design of a robust adaptive fuzzy estimator for a drive system with a flexible joint is presented. The proposed estimator ensures variable Kalman gain (based on the Mahalanobis distance) as well as the estimation of the system parameters (based on the fuzzy system). The obtained value of the time constant of the load machine is used to change the values in the system state matrix and to retune the parameters of the state controller. The proposed control structure (fuzzy Kalman filter and adaptive state controller) is investigated in simulation and experimental tests.

The paper presents a general concept of geodetic observations adjustment based on application of the Edgeworth' series and the principle of an alternative choice. The Edgeworth' series approximates the empirical distribution of measurement errors and gives an opportunity to modify the empirical characteristics of errors distribution. The method of estimation used is based on the principle of the alternative choice. Its natural robustness for outliers was the basis for newly created method called PAC-E. The paper presents the algorithm and some numerical tests that were carried out to compare the results of the proposed method with the results of the classical LS adjustment. Special attention was paid on the effect of non-zero excess and robustness of the proposed method.

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.}

A method of the improvement of the total station observations 3D adjustment by using precise geoid model is presented. The novel concept of using the plumb line direction obtained from the precise geoid model in combined GPS/total station data adjustment is applied. It is concluded that results of the adjustment can be improved if data on plumb line direction is used. Theoretical background shown in the paper was proved with an experiment based on the total station and GPS measurements referred to GRS80 geocentric reference system and with the use of GUGIK2001 geoid model for Poland.

We present a summary of research carried out in 2019–2022 in Poland in the area of general theory and methodology in geodesy. The study contains a description of original contributions by authors affiliated with Polish scientific institutions. It forms part of the national report presented at the 28th General Assembly of the International Union of Geodesy and Geophysics (IUGG) taking place on 11-20 July 2023 in Berlin, Germany. The Polish authors developed their research in the following thematic areas: robust estimation and its applications, prediction problems, cartographic projections, datum transformation problems and geometric geodesy algorithms, optimization and design of geodetic networks, geodetic time series analysis, relativistic effects in GNSS (Global Navigation Satellite System) and precise orbit determination of GNSS satellites. Much has been done on the subject of estimating the reliability of existing algorithms, but also improving them or studying relativistic effects. These studies are a continuation of work carried out over the years, but also they point to new developments in both surveying and geodesy.We hope that the general theory and methodology will continue to be so enthusiastically developed by Polish authors because although it is not an official pillar of geodesy, it is widely applicable to all three pillars of geodesy.