Robust estimation of deformation from observation differences for free control networks

Abstract

Deformation measurements have a repeatable nature. This means that deformation measurements are performed often with the same equipment, methods, geometric conditions and in a similar environment in epochs 1 and 2 (e.g., a fully automated, continuous control measurements). It is, therefore, reasonable to assume that the results of deformation measurements can be distorted by both random errors and by some non-random errors, which are constant in both epochs. In other words, there is a high probability that the difference in the accuracy and precision of measurement of the same geometric element of the network in both epochs has a constant value and sign. The constant errors are understood, but the manifestation of these errors is difficult to determine in practice. For free control networks (the group of potential reference points in absolute control networks or the group of potential stable points in relative networks), the results of deformation measurements are most often processed using robust methods. Classical robust methods do not completely eliminate the effect of constant errors. This paper proposes a new robust alternative method called REDOD. The performed tests showed that if the results of deformation measurements were additionally distorted by constant errors, the REDOD method completely eliminated their effect from deformation analysis results. If the results of deformation measurements are only distorted by random errors, the REDOD method yields very similar deformation analysis results as the classical IWST method. The numerical tests were preceded by a theoretical part. The theoretical part K. Nowel (B) · W. Kamiński Institute of Geodesy, University of Warmia and Mazury in Olsztyn, 1 Oczapowskiego Str., 10-719 Olsztyn, Poland e-mail: krzysztof.nowel@uwm.edu.pl W. Kamiński e-mail: waldemar.kaminski@uwm.edu.pl describes the algorithm of classical robust methods. Particular attention was paid to the IWST method. In relation to classical robust methods, the optimization problem of the new REDOD method was formulated and the algorithm for its solution was derived.

Geometric deformation analysis in free geodetic networks: case study for Fruska Gora in Serbia

“…The best known methods from this group are the IWST method, which satisfies the condition of minimum sum of displacement components module (Chen, 1983;Setan and Singh, 2001), which differs only by weight function (L1-norm function for the displacement vector components, L1-norm function for the displacement lengths and Huber function for the displacement vector components (Caspary and Borutta, 1987;Setan andSingh, 2001, Nowel andKamiński, 2014). Robust methods are based on the S transformation (Helmert's similarity transformation):…”

“…If ≤ , , , the point is stable, otherwise ( > , , ) the point is unstable. Detailed explanations concerning the robust methods can be found in contributions of the following authors: (Chen, 1983;Caspary and Borutta, 1987;Setan and Singh, 2001;Nowel and Kamiński, 2014;Nowel, 2015Nowel, , 2016aNowel, , 2016b.…”

“…IWST and LAS are based on the S transformation for the purpose of detecting the movement trend of the points. The robust estimation of deformation from observation differences (REDOD and its generalization GREDOD) is especially recommended when the observation plan is completely the same for both measurement epochs in order to keep constant possible non-random errors for one geometric network element in both epochs under control (Nowel and Kamiński, 2014;Nowel, 2015). In the literature one can find applications in deformation analyses of the R-estimation (Duchnowski, 2009 and2013), as well as of the M split estimation (Wiśniewski, 2009;Zienkiewicz, 2014;Zienkiewicz and Baryla, 2015;Zienkiewicz et al, 2017).…”

Theoretical geodesy

“…The existence of constant errors reduces the quality of deformations analysis. To eliminate effects of those errors in deformation analysis Nowel and Kamiński (2014) proposed a new robust alternative method called robust estimation of deformation from observation differences (REDOD). This method is based on the assumption, that the vector d is a function of differences between independent observations, ) (…”

Subjective Breakdown Points of R-estimators Applied in Deformation Analysis

“…On the other hand, such knowledge might also have practical importance, for example, when one can predict the locations of outliers. There are several robust methods which are applied in geodesy of surveying, like for example, robust M-estimation (with several different methods), M split estimation or R-estimation (for example, Xu 1989Xu , 2005Yang 1994;Kargoll 2005;Duchnowski 2010;Wiśniewski 2009Wiśniewski , 2014Nowel, Kamiński 2014). In the present paper, we will focus on R-estimates which can be applied, for example, in deformation analysis (Duchnowski 2009(Duchnowski , 2010(Duchnowski , 2013Kargoll 2005).…”