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 justified for the linear functions. In case of the non-linear functions, the first-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 simplification. 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 confirm 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.
Anderson, T.V. and Mattson, C.A. (2012). Propagating skewness and kurtosis through engineering models for low-cost, meaningful, nondeterministic design. J Mech Design, 134(10). DOI: 10.1115/1.4007389.
Del Moral, P., Doucet, A. and Jasra, A. (2006). Sequential Monte Carlo samplers. J R Stat Soc B, 68(3), 411-436. DOI: 10.1111/j.1467-9868.2006.00553.x.
Duchnowski, R. and Wiśniewski, Z. (2014). Comparison of two unconventional methods of estimation applied to determine network point displacement. Surv Rev, 46(339), 401-405. DOI: 10.1179/1752270614Y.0000000127.
Duchnowski, R. and Wiśniewski, Z. (2017). Accuracy of the Hodges-Lehmann estimates computed by applying Monte Carlo simulations. Acta Geod Geophys. DOI: 10.1007/s40328-016-0186-0.
Duchnowski, R. and Wyszkowska, P. (2017). Leptokurtosis of error distribution and its influence on estimation accuracy. The case of three estimates applied in adjustment of geodetic measurements. In: Baltic Geodetic Congress, June 22-25 2017, IEEE. DOI: 10.1109/BGC.Geomatics.2017.32.
Eckhardt, R. (1987). Stan Ulam, John von Neumann and the Monte Carlo method. Los Alamos Sci, 15, 131-137.
Fishman, G. S. (1986). A Monte Carlo sampling plan for estimating network reliability. Oper Res, 34(4), 581-594. DOI: 10.1287/opre.34.4.581.
Hekimoglu, S. and Berber, M. (2003). Effectiveness of robust methods in heterogeneous linear models. J Geod, 76(11), 706-713. DOI: 10.1007/s00190-002-0289-y.
JCGM 101:2008. (2008). Evaluation of measurement data - Supplement 1 to the “Guide to the expression of uncertainty in measurement” - Propagation of distributions using a Monte Carlo method. Geneva: Joint Committee for Guides in Metrology, International Organization for Standardization.
Kass, R.E., Eden, U. and Brown, E. (2014). Analysis of Neural Data. New York: Springer. DOI: 10.1007/978-1-4614-9602-1.
Lee, S. H. and Chen, W. (2009). A comparative study of uncertainty propagation methods for black-boxtype problems. Struct Multidiscip O, 37(3), 239-253. DOI: 10.1007/s00158-008-0234-7.
Ligas, M. (2013). Simple solution to the three point resection problem. J Surv Eng, 139(3), 120-125. DOI: 10.1061/(ASCE)SU.1943-5428.0000104.
Mikhail, E.M. and Ackermann, F.E. (1976). Observations and least squares. New York, NY: Harper and Row.
Metropolis, N. (1987). The beginning of the Monte Carlo method. Los Alamos Sci, 15, 125-130.
Metropolis, N. and Ulam, S. (1949). The Monte Carlo method. J Am Stat Assoc, 44(247), 335-341. DOI: 10.1080/01621459.1949.10483310.
Oehlert, G.W. (1992). A note on the delta method. Am Stat, 46(1), 27-29. DOI: 10.1080/00031305.1992.10475842.
Ramaley, J. F. (1969). Buffon’s noodle problem. Am Math Mon, 76(8), 916-918. DOI: 10.2307/2317945.
Xu, P. (2005). Sign-constrained robust least squares, subjective breakdown point and the effect of weights of observations on robustness. J Geod, 79(1), 146-159. DOI: 10.1007/s00190-005-0454-1.
Wang, Y. (2011). Quantum Monte Carlo simulation. Ann Appl Stat 5(2A), 669-683. DOI: 10.1214/10-AOAS406.
Warnock, T. (1987). Random-number generators. Los Alamos Sci, 15, 137-141.
Wyszkowska, P. and Duchnowski, R. (2017). Subjective breakdown points of R-estimators applied in deformation analysis. In: Environmental Engineering 10th International Conference, April 27-28 2017, (pp. 1-6). DOI: 10.3846/enviro.2017.250.