The paper presents an empirical comparison of performance of three well known M -estimators (i.e. Huber, Tukey and Hampel's M -estimators) and also some new ones. The new M -estimators were motivated by weighting functions applied in orthogonal polynomials theory, kernel density estimation as well as one derived from Wigner semicircle probability distribution. M -estimators were used to detect outlying observations in contaminated datasets. Calculations were performed using iteratively reweighted least-squares (IRLS). Since the residual variance (used in covariance matrices construction) is not a robust measure of scale the tests employed also robust measures i.e. interquartile range and normalized median absolute deviation. The methods were tested on a simple leveling network in a large number of variants showing bad and good sides of M -estimation. The new M -estimators have been equipped with theoretical tuning constants to obtain 95% effi ciency with respect to the standard normal distribution. The need for data -dependent tuning constants rather than those established theoretically is also pointed out.

We present a method of approximation of a deformation eld based on the local a ne transformations constructed based on n nearest neighbors with respect to points of adopted grid. The local a ne transformations are weighted by means of inverse distance squared between each grid point and observed points (nearest neighbors). This work uses a deformation gradient, although it is possible to use a displacement gradient instead -the two approaches are equivalent. To decompose the deformation gradient into components related to rigid motions (rotations, translations are excluded from the deformation gradient through di erentiation process) and deformations, we used a polar decomposition and decomposition into a sum of symmetric and an anti-symmetric matrices (tensors). We discuss the results from both decompositions. Calibration of a local a ne transformations model (i.e., number of nearest neighbors) is performed on observed points and is carried out in a cross-validation procedure. Veri cation of the method was conducted on simulated data-grids subjected to known (functionally generated) deformations, hence, known in every point of a study area.

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