Propagation of uncertainty by Monte Carlo simulations in case of basic geodetic computations

Abstract

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.

Testing normality of chosen R-estimates used in deformation analysis

“…What is more, the worse results (the bigger di erences between theoretical and empirical values) are acquired for coordinate X1, contrary to Variant IV. This results from the propagation of the expected values from angles and distances to coordinates for nonlinear functions (intersections and resection) and when accuracies of measurements are much di erent, see for example (Wyszkowska, 2017).…”

Monte-Carlo-based uncertainty propagation in the context of Gauss–Markov model: a case study in coordinate transformation

“…In that case, the first order of the Taylor series expansion can be applied to solve it. Nevertheless, the linearized model can often provide an inadequate representation of the uncertainties of parameters when the model itself is highly non-linear and/or the uncertainty of measurements is very low [19]. Analytically expressible solutions are ideal only in cases where they do not introduce any approximation.…”