ROBUSTNA DOLOČITEV PRIBLIŽNIH KOORDINAT V HORIZONTALNI GEODETSKI MREŽI
ROBUST DETERMINATION OF APPROXIMATE COORDINATES IN A HORIZONTAL GEODETIC NETWORK

Sandi Berk

DOI: 10.15292/geodetski-vestnik.2010.01.009-030

 

Izvleček:

Predstavljen je postopek robustne določitve približnih koordinat točk v horizontalni geodetski mreži. Primarni cilj ni natančnost, ampak čim večja zanesljivost pridobljenih koordinat. Razlog določitve je omogočiti nadaljnjo obdelavo po modelu Gauß-Markova. Ker gre za linearen matematični model, sodijo približne vrednosti neznank med nujno potrebne vhodne podatke. Obravnavane so razpoložljive metode določitve koordinat točk in kombinatorika predoločenosti rešitev v odvisnosti od nadštevilnih opazovanj. Uporabljen je geometrijski pristop k reševanju, ki sloni na dejstvu, da vsako točko mreže določa presek para krivulj. Geometrijska kakovost posamezne rešitve je ovrednotena na podlagi presečnega kota obeh krivulj. Posamezni rešitvi dodeljena utež je funkcija tega kota. Izračun koordinat je postopen; v vsakem koraku sta določeni koordinati ene nove točke. Algoritem vključuje iskanje tipične rešitve med vsemi rešitvami za posamezno točko. Zaradi izogibanja morebitnim grobim pogreškom temelji postopek na uporabi robustne statistike. Na praktičnem primeru je preizkušena učinkovitost treh osnovnih mer lokacije. Gre za posplošitve povprečja, mediane in modusa – težiščno, središčno in gostiščno točko. Grobi pogreški so v mrežo uvedeni z uporabo simulacije Monte Carlo.

Ključne besede: horizontalna geodetska mreža, mere lokacije, približne koordinate, robustna statistika, slabo pogojene rešitve, večvariantni modus

 

Abstract:

A procedure of robust determination of the approximate coordinates of points in a horizontal geodetic network is presented. The primary aim is not accuracy but the reliability of the obtained coordinates. The motivation is to enable further processing according to the Gauss-Markov model. It is a linear mathematical model, thus the approximate values of unknowns are necessary input data. Disposable methods of determining coordinates of points and combinatorics of over-determined solutions dependent upon redundant observations are discussed. A geometrical approach is used, based on the fact that every point is an intersection of a pair of curves. The geometrical quality of each individual solution is evaluated from the intersection angle of both curves. A weight assigned to each solution is a function of that angle. Calculating coordinates is a successive procedure; each step assures the determination of one network point. The algorithm comprises searching for a typical solution among all solutions for an individual point. In order to avoid eventual gross errors, the procedure is based on robust statistics. The efficiency of three basic measures of location is tested on a practical example. The point is a generalization of mean, median, and mode i.e. the centre of mass, spatial median, and spatial mode. Gross errors are introduced into the network by using a Monte Carlo simulation.

Keywords: approximate coordinates, horizontal geodetic network, measures of location, multivariate mode, poor solutions, robust statistics

 

Literatura / References:

Bajaj, C. (1988). The Algebraic Degree of Geometric Optimization Problems. Discrete & Computational Geometry, 3(1), 177–191.
http://dx.doi.org/10.1007/BF02187906

Berk, S., Janežič, M. (1995). TRIM – program za izravnavo triangulacijskih mrež. Geodetski vestnik, 39(4), 271–279.

Berk, S. (1996). Izravnava in statistična analiza temeljnih horizontalnih geodetskih mrež. Diplomska naloga. Ljubljana: Fakulteta za gradbeništvo in geodezijo.

Berk, S. (2008). Programski paket TRIM: TRIM Izračuni, različica 1.0. Uporabniški priročnik.

Caspary, W. F. (1988). Concepts of Network and Deformation Analysis. 2. (popravljeni) ponatis. Kensington: The University of New South Wales.

Chandrasekaran, R., Tamir, A. (1990). Algebraic Optimization: The Fermat-Weber Location Problem. Mathematical Programming, 46(1–3), 219–224.
http://dx.doi.org/10.1007/BF01585739

Grigillo, D., Stopar, B. (2003). Metode odkrivanja grobih pogreškov v geodetskih opazovanjih. Geodetski vestnik, 47(4), 387–403.

Kärkkäinen, T., Äyrämö, S. (2005). On Computation of Spatial Median for Robust Data Mining. Proceedings of the 6th Conference on Evolutionary and Deterministic Methods for Design, Optimisation and Control with Applications to Industrial and Societal Problems (EUROGEN 2005). Technische Universität München.

Košmelj, B., Arh, F., Doberšek - Urbanc, A., Ferligoj, A., Omladič, M. (1993). Statistični terminološki slovar. 1. izdaja. Ljubljana: Statistično društvo Slovenije in Društvo matematikov, fizikov in astronomov Slovenije.

Lopuhaä, H. P., Rousseeuw, P. J. (1991). Breakdown Points of Affine Equivariant Estimators of Multivariate Location and Covariance Matrices. The Annals of Statistics, 19(1), 229–248.
http://dx.doi.org/10.1214/aos/1176347978

Mihailović, K. (1992). Geodezija: Izravnanje geodetskih mreža. Beograd: Naučna knjiga in Građevinski fakultet.

Milasevic, P., Ducharme, G. R. (1987). Uniqueness of the Spatial Median. The Annals of Statistics, 15(3), 1332–1333.
http://dx.doi.org/10.1214/aos/1176350511

Mishra, S. K. (2004). Median as a Weihted Arithmetic Mean of All Sample Observations. Economics Bulletin, 3(18), 1–6.

Mouratidis, K., Papadias, D., Papadimitriou, S. (2005). Medoid Queries in Large Spatial Databases. Proceedings of the 9th International Symposium on Spatial and Temporal Databases (Lecture Notes in Computer Science): Advances in Spatial and Temporal Databases. New York: Springer-Verlag.

Press, W. H., Teukolsky, S. A., Vetterling, W. T., Flannery, B. P. (1992). Numerical Recipies in C: The Art of Scientific Computing. 2. izdaja. Cambridge University Press.

Sager, T. W. (1979). An Iterative Method for Estimating a Multivariate Mode and Isopleth. Journal of the American Statistical Association, 74(366), 329–339.
http://dx.doi.org /10.1080/01621459.1979.10482514

Smoltczyk, U. (2002). Geotechnical Engineering Handbook. Fundamentals. Berlin