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Geodesy and Cartography

The Journal of Committee on Geodesy of Polish Academy of Sciences

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Kriging approach for local height transformations

Marcin Ligas
  • Corresponding author
  • AGH University of Science and Technology Faculty of Mining Surveying and Environmental Engineering Department of Geomatics, 30 Mickiewicza Al., 30-059 Krakow, Poland
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/ Marek Kulczycki
  • AGH University of Science and Technology Faculty of Mining Surveying and Environmental Engineering Department of Geomatics, 30 Mickiewicza Al., 30-059 Krakow, Poland
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Published Online: 2014-06-13 | DOI: https://doi.org/10.2478/geocart-2014-0002


In the paper a transformation between two height datums (Kronstadt’60 and Kronstadt’86, the latter being a part of the present National Spatial Reference System in Poland) with the use of geostatistical method - kriging is presented. As the height differences between the two datums reveal visible trend a natural decision is to use the kind of kriging method that takes into account nonstationarity in the average behavior of the spatial process (height differences between the two datums). Hence, two methods were applied: hybrid technique (a method combining Trend Surface Analysis with ordinary kriging on least squares residuals) and universal kriging. The background of the two methods has been presented. The two methods were compared with respect to the prediction capabilities in a process of crossvalidation and additionally they were compared to the results obtained by applying a polynomial regression transformation model. The results obtained within this study prove that the structure hidden in the residual part of the model and used in kriging methods may improve prediction capabilities of the transformation model.


W artykule przedstawiono lokalną transformację między dwoma układami wysokości (Kronsztadt’60 oraz Kronsztadt’86, ostatni z nich będący obecnie częścią Państwowego Systemu Odniesień Przestrzennych w Polsce) z wykorzystaniem metod geostatystycznych - kriging. Ze względu na fakt, iż różnice wysokości między dwoma układami na punktach dostosowania wykazywały silny trend pod uwagę wzięto tylko te metody, które uwzględniają tego typu niestacjonarność procesu. Zastosowano dwie metody: hybrydową (Analiza Trendu Powierzchniowego z interpolacją reszt do modelu za pomocą krigingu zwyczajnego) oraz kriging uniwersalny. Przedstawiono rys teoretyczny obydwu metod. Dokonano porównania wyżej wymienionych metod pod względem ich zdolności predykcyjnych w procesie kroswalidacji modeli a zarazem otrzymane wyniki skonfrontowano z wynikami otrzymanymi z regresji wielomianowej. Otrzymane wyniki dowodzą, iż struktura ukryta w rezydualnej części modelu używana przez kriging może podnieść zdolności predykcyjne modelu transformacji

Keywords: kriging; polynomial regression; height transformation


  • Armstrong, M. (1998). Basic linear geostatistics. Berlin: Springer.Google Scholar

  • Chauvet, P. & Galli A. (1982). Universal kriging. Cours - C-96, Centre de Geostatistique, Ecole des Mines de Paris.Google Scholar

  • Chiles, J.P. & Delfi ner P. (1999). Geostatistics - modeling spatial uncertainty. New York: John Wiley & Sons.Google Scholar

  • Cressie, N.A.C. (1990). The origins of kriging. Mathematical Geology, 22(3), 239-252.CrossrefGoogle Scholar

  • Cressie, N.A.C. (1993). Statistics for spatial data. New York: John Wiley & Sons.Google Scholar

  • Dermanis, A. (1984). Kriging and collocation - a comparison. Manuscripta geodaetica, 9, 159-167.Google Scholar

  • Deutsch, C.V. (1996). Correcting for negative weights in ordinary kriging. Computers and Geosciences, 22(7), 765-773.Google Scholar

  • Efron, B. & Tibshirani, R.J. (1993). An introduction to the Bootstrap. New York: Chapman&Hall/CRC.Google Scholar

  • Goldberger, A.S. (1962). Best linear unbiased prediction in the generalized linear regression models. Journal of the American Statistical Association, 57(298), 369-375.Google Scholar

  • Goovaerts, P. (1997). Geostatistics for natural resources evaluation. New York: Oxford University Press.Google Scholar

  • Hastie, T., Tibshirani R. & Friedman J. (2009). The elements of statistical learning - data mining, inference, and prediction. Berlin: Springer.Google Scholar

  • Heiskanen, A.W. & Moritz H. (1967). Physical Geodesy. San Francisco W. H. Freeman and Company.Google Scholar

  • Krarup, T. (1969). A contribution to the mathematical foundation of physical geodesy. Geodaetisk Institut, Kobenhavn.Google Scholar

  • Ligas, M. & Banasik, P. (2012). Local height transformation through polynomial regression. Geodesy and Cartography, 61(1), 3-17. DOI: 10.2478/v10277-012-0018-5.CrossrefGoogle Scholar

  • Ligas, M. & Kulczycki, M. (2010). Simple spatial prediction - least squares prediction, simple kriging, and conditional expectation of normal vector. Geodesy and Cartography, 59(2), 69-81.Google Scholar

  • Maddala, G. S. (1992). Introduction to econometrics. New York: MacMillan Publishing Company.Google Scholar

  • Moritz, H. (1980). Advanced physical geodesy. Karlsruhe: Herbert Wichmann Verlag.Google Scholar

  • Olea, R .A. (1999). Geostatistics for engineers and earth scientists. Boston: Kluwer Academic Publishers.Google Scholar

  • Olea, R. A. (2006). A six-step practical approach to semivariogram modeling. Stoch Environ Res Risk Assess, 20, 307-318.Google Scholar

  • RMAiC. (2012). Regulation of the Minister of Administration and Digitization of 14 February 2012 concerning geodetic, gravimetric and magnetic networks; [Rozporządzenie Ministra Administracji i Cyfryzacji z dnia 14 lutego 2012 r. w sprawie osnów geodezyjnych, grawimetrycznych i magnetycznych. Dz.U. 2012 nr 0 poz. 352] Google Scholar

  • Sanso, F. Statistical analysis of environmental data. Lecture notes available at http://geomatica.como.polimi.it/corsi/ Google Scholar

  • Schabenberger, O., Gotway, C.A. (2005). Statistical methods for spatial data analysis. New York: Chapman & Hall/CRC.Google Scholar

  • Stein, L.M. (1999). Interpolation of spatial data - some theory for kriging. New York: Springer.Google Scholar

  • Wackernagel, H. (1995). Multivariate geostatistics. Berlin: Springer. Google Scholar

About the article

Received: 2013-08-30

Accepted: 2014-03-03

Published Online: 2014-06-13

Published in Print: 2014-06-01

Citation Information: Geodesy and Cartography, Volume 63, Issue 1, Pages 25–37, ISSN (Online) 2300-2581, ISSN (Print) 2080-6736, DOI: https://doi.org/10.2478/geocart-2014-0002.

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© by Marcin Ligas. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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