Mathematical Geosciences

, Volume 47, Issue 7, pp 867–883 | Cite as

Correction of Gravimetric Geoid Using Symbolic Regression

  • B. Paláncz
  • J. L. Awange
  • L. Völgyesi


In this study, the problem of geoid correction based on GPS ellipsoidal height measurements is solved via symbolic regression (SR). In this case, when the quality of the approximation is overriding, SR employing Keijzer expansion to generate initial trial function population can supersede traditional techniques, such as parametric models and soft computing models (e.g., artificial neural network approach with different activation functions). To demonstrate these features, numerical computations for correction of the Hungarian geoid have been carried out using the DataModeler package of Mathematica. Although the proposed SR method could reduce the average error to a level of 1–2 cm, it has two handicaps. The first one is the required high computation power, which can be eased by the employment of parallel computation via multicore processor. The second one is the proper selection of the initial population of the trial functions. This problem may be solved via intelligent generation technique of this population (e.g., Keijzer-expansion).


Symbolic regression Genetic programming Artificial neural networks Pareto optimality Geoid GPS 



The authors thank the Editor and the Reviewers for their comments and suggestions during the review period of the manuscript, but take the responsibility of any errors. This is a TIGeR publication No. 583. This research was funded partially by OTKA project No. 76231.


  1. Babu BV, Karthik (2007) Genetic Programming for symbolic regression of chemical process systems, Engineering Letters 14:2, EL-14 2 6 (advanced on line publication)Google Scholar
  2. Banks C (2002) Searching for Lyapunov functions using genetic programming, Technical report, Virginia Polytechnic Institute and State University, BlacksburgGoogle Scholar
  3. Cramer NL (1985) A representation for the adaptive generation of simple sequential programs. In: Grefenstette JJ (ed) Proceedings of the 1st International Conference on Genetic Algorithms and Their Applications, Erlbaum, pp 183–187Google Scholar
  4. Davidson JW, Savic DA, Walters GA (2003) Symbolic and numerical regression: experiments and applications. Inf Sci 150(12):95–117CrossRefGoogle Scholar
  5. Danila U (2006) Corrective surface for GPS-levelling in Moldova, Master’s of Sci. Thesis, Royal Institute of Technology (KTH) School of Architecture and the Built Environment, TRITA-GIT EX 06–001 Geodesy Report No. 3089, Stockholm, SwedenGoogle Scholar
  6. Duquenne H, Jiang Z, Lemarie C (1995) Geoid determination and levelling by GPS: some experiments on the test network. IAG Symposia Gravity and Geoid, 113. Springer, pp 559–568Google Scholar
  7. Featherstone W (2000) Refinement of gravimetric geoid using GPS and levelling data. J Surv Eng 126(2): 27–56CrossRefGoogle Scholar
  8. Ferreira C (2006) Gene expression programming: Mathematical modeling by an artificial intelligence, 2nd edn. Springer, BerlinGoogle Scholar
  9. Fotopoulos G (2005) Calibration of geoid error models via a combined adjustment of ellipsoidal, orthometric and gravimetric geoid height data. J Geod 79(1–3):111–123CrossRefGoogle Scholar
  10. Fotopoulos G, Sideris MG (2005) Spatial modeling and analysis of adjusted residuals over a network of GPS-levelling bench marks. Geomatica 59(3):251–262Google Scholar
  11. Garg A, Tai K (2011) A hybrid genetic programming—artificial neural network approach for modeling of vibratory finishing process, 2011 International Conference on Information and Intelligent Computing IPCSIT vol 18. IACSIT Press, Singapore, pp 14–19Google Scholar
  12. Heiskanen W, Moritz H (1967) Physical Geodesy. W H Freeman and Co., San FranciscoGoogle Scholar
  13. Iliffe JC, Ziebart M, Cross PA, Forsberg R, Strykowski G, Tscherning CC (2003) OGSM02: a new model for converting GPS-derived heights to local height datums in Great Britain and Ireland. Surv Rev 37(290):276–293CrossRefGoogle Scholar
  14. Kavzoglu T, Saka MH (2005) Modelling local GPS/levelling geoid undulations using artificial neural networks. J Geod 78:520–527CrossRefGoogle Scholar
  15. Kecman V (2001) Learning and soft computing: support vector machines, neural networks, and fuzzy logic models (complex adaptive systems). The MIT Press, CambridgeGoogle Scholar
  16. Keijzer M (2003) Regression with interval arithmetic and linear scaling. In: Genetic Programming, 6th European Conference, EuroGP 2003, vol 2610. Springer, pp 70–82Google Scholar
  17. Kenyeres A, Virág G (1998) Testing recent geoid models with GPS/levelling in Hungary. Rep Finn Geod Inst Masal 98(4):217–223Google Scholar
  18. Kotsakis C, Fotopulos G, Sideris MG (2001) Optimal fitting of gravimetric geoid undulations to GPS/levelling data using an extended similarity transformation model. In: The 27th Annual Meeting of the Canadian Geophysical Union, Ottawa, CanadaGoogle Scholar
  19. Koza JR (1992) Genetic programming: on the programming of computers by means of natural selection. The MIT Press, CambridgeGoogle Scholar
  20. Kwon YK, Moon BR (2005) Critical heat flux function approximation using genetic algorithms. IEEE Trans Nuclear Sci 52(2):535–545CrossRefGoogle Scholar
  21. Langdon WB, Gustafson SM (2010) Geneteic programming and evolvable machines: 10 years of reviews. Genet Program Evolvable Mach 11:321–338CrossRefGoogle Scholar
  22. Lin Lao-Sheng (2007) Application of a back-propagation artificial neural network to regional grid-based geoid model generation using gps and levelling data. J Surv Eng 133(2):81–89CrossRefGoogle Scholar
  23. Liu XG, Wu XP, Wang K (2012) Construction of least squares collocation models for single component and composite components of disturbed gravity gradient. Chin J Geophys 55(2):294–302CrossRefGoogle Scholar
  24. Morales CO (2004) Symbolic regression problems by genetic programming with multi-branches. MICAI 2004: Advances in Artificial Intelligence, pp 717–726Google Scholar
  25. Nahavandchi H, Soltanpour A (2004) An attempt to define a new height datum in Norvay. The Geodesy and Hydrography Days, 4–5 Nov. Sandnes, NorwayGoogle Scholar
  26. Paláncz B, Awange JL (2012) Application of Pareto optimality to linear models with errors-in-all-variables. J Geod 86(7):531–545CrossRefGoogle Scholar
  27. Parasuraman K, Elshorbagy A, Carey SK (2007) Modelling the dynamics of the evapotranspiration process using genetic programming. Hydrol Sci J 52(3):563–578. doi: 10.1623/hysj.52.3.563 CrossRefGoogle Scholar
  28. Santini M, Tettamanzi A (2001) Genetic Programming for financial time series prediction. In: Euro GPO’01 Proceedings, Lectures Notes in Computer Science 2038, Genetic Programming, pp 361–371Google Scholar
  29. Schmidt M, Lipson H (2009) Distilling free-form natural laws from experimental data. Science 324:81–85Google Scholar
  30. Smits G, Kotanchek M (2004) Pareto-front exploitation in symbolic regression. In: Genetic Programming Theory and Practice II. Springer, Ann Arbor USA, pp 283–299Google Scholar
  31. Soltanpour A, Nahavandchi H, Featherstone WE (2006) Geoid-type surface determination using wavelet-based combination of gravimetric quasi/geoid and GPS/levelling data. Geophys Res Abstr 8:4612Google Scholar
  32. Wu CH, Chou HJ, Su WH (2007) A genetic approach for coordinate transformation test of gps positioning. IEEE Geosci Remote Sens Lett 4(2):297–301CrossRefGoogle Scholar
  33. Wu CH, Chou HJ, Su WH (2008) Direct transformation of coordinates for GPS positioning using techniques of genetic programming and symbolic regression on partitioned data. Eng Appl Artif Intell 21:1347–1359CrossRefGoogle Scholar
  34. Wu CH, Su WH (2013) Lattice-based clustering and genetic programming for coordinate transformation in GPS applications. Comput Geosci 52:85–94CrossRefGoogle Scholar
  35. Zaletnyik P, Paláncz B, Völgyesi L, Kenyeres A (2007) Correction of the gravimetric geoid using GPS leveling data. Geomatikai Közlemények, vol X, pp 231–240 (In Hungarian)Google Scholar
  36. Zaletnyik P, Völgyesi L, Paláncz B (2008) Modelling local GPS/leveling geoid undulations using support vector machines. Period Polytech Civ Eng 52(1):39–43CrossRefGoogle Scholar

Copyright information

© International Association for Mathematical Geosciences 2015

Authors and Affiliations

  1. 1.Department of Photogrammetry and GeoinformaticsBudapest University of Technology and EconomicsBudapestHungary
  2. 2.Western Australian Centre for Geodesy and the Institute for Geoscience ResearchCurtin UniversityPerthAustralia
  3. 3.Department of Geodesy and SurveyingBudapest University of Technology and EconomicsBudapestHungary

Personalised recommendations