Mathematical Geosciences

, Volume 45, Issue 3, pp 297–319 | Cite as

A Robust Multiquadric Method for Digital Elevation Model Construction

  • Chuanfa ChenEmail author
  • Yanyan Li


The multiquadric method (MQ) with high interpolation accuracy has been widely used for interpolating spatial data. However, MQ is an exact interpolation method, which is improper to interpolate noisy sampling data. Although the least squares MQ (LSMQ) has the ability to smooth out sampling errors, it is inherently not robust to outliers due to the least squares criterion in estimating the weights of sampling knots. In order to reduce the impact of outliers on the accuracy of digital elevation models (DEMs), a robust method of MQ (MQ-R) has been developed. MQ-R includes two independent procedures: knot selection and the solution of the system of linear equations. The two independent procedures were respectively achieved by the space-filling design and the least absolute deviation, both of which are very robust to outliers. Gaussian synthetic surface, which is subject to a series of errors with different distributions, was employed to compare the performance of MQ-R with that of LSMQ. Results indicate that LSMQ is seriously affected by outliers, whereas MQ-R performs well in resisting outliers, and can construct satisfactory surfaces even though the data are contaminated by severe outliers. A real-world example of DEM construction was employed to evaluate the robustness of MQ-R, LSMQ, and the classical interpolation methods including inverse distance weighting method, thin plate spline, and ANUDEM. Results showed that compared with the classical methods, MQ-R has the highest accuracy in terms of root mean square error. In conclusion, when sampling data is subject to outliers, MQ-R can be considered as an alternative method for DEM construction.


Robust Interpolation Radial basis function Digital elevation model 



This work is supported by National Natural Science Foundation of China (Grant No. 41101433), by Young and Middle-Aged Scientists Research Awards Fund of Shangdong Province (Grant No. BS2012HZ010), by the Key Laboratory of Surveying and Mapping Technology on Island and Reef, National Administration of Surveying, Mapping and Geoinfomation (Grant No. 2011B10), by the Key Laboratory of Coastal Zone Environmental Processes, YICCAS (Grant No. 201209), and by the Special Project Fund of Taishan Scholars of Shandong Province.


  1. Aguilar FJ, Aguera F, Aguilar MA, Carvajal F (2005) Effects of terrain morphology, sampling density, and interpolation methods on grid DEM accuracy. Photogramm Eng Remote Sens 71(7):805–816 Google Scholar
  2. Arrell K, Wise S, Wood J, Donoghue D (2008) Spectral filtering as a method of visualising and removing striped artefacts in digital elevation data. Earth Surf Process Landf 33(6):943–961 CrossRefGoogle Scholar
  3. Bjorck A (1968) Iterative refinement of linear least squares solutions II. BIT Numer Math 8(1):8–30 CrossRefGoogle Scholar
  4. Bjorck A (1996) Numerical methods for least squares problems. SIAM, Philadelphia CrossRefGoogle Scholar
  5. Brus DJ, de Gruijter JJ (1997) Random sampling or geostatistical modelling? Choosing between design-based and model-based sampling strategies for soil (with discussion). Geoderma 80(1–2):1–44 CrossRefGoogle Scholar
  6. Brus DJ, de Gruijter JJ, van Groenigen JW (2007) Designing spatial coverage samples using the k-means clustering algorithm. In: Lagacherie P, McBratney AB, Voltz M (eds) Digital soil mapping: an introductory perspective. Elsevier, Amsterdam, pp 183–192 Google Scholar
  7. Carlson RE, Foley TA (1991) The parameter R 2 in multiquadric interpolation. Comput Math Appl 21(9):29–42 CrossRefGoogle Scholar
  8. Chen S, Cowan C, Grant P (1991) Orthogonal least squares learning algorithm for radial basis function networks. IEEE Trans Neural Netw 2(2):302–309 CrossRefGoogle Scholar
  9. Chen CF, Li YY, Yue TX (2013) Surface modeling of DEMs based on a sequential adjustment method. Int J Geogr Inf Sci. doi: 10.1080/13658816.2012.704037 Google Scholar
  10. Chen CF, Yue TX, Li YY (2012a) A high speed method of SMTS. Comput Geosci 41:64–71 CrossRefGoogle Scholar
  11. Chen CF, Fan ZM, Yue TX, Dai HL (2012b) A robust estimator for the accuracy assessment of remote-sensing-derived DEMs. Int J Remote Sens 33(8):2482–2497 CrossRefGoogle Scholar
  12. Cochran WG (1977) Sampling techniques. Wiley, New York Google Scholar
  13. Cormen TH, Leiserson CE, Rivest RL, Stein C (2009) Introduction to algorithms. MIT Press, Cambridge Google Scholar
  14. Cressie NAC (1993) Statistics for spatial data. Wiley, New York Google Scholar
  15. Delmelle E (2009) Spatial sampling. In: Rogerson P, Fotheringham S (eds) Handbook of spatial analysis. Sage Publication, London, UK, 528 pp Google Scholar
  16. Delmelle EM, Goovaerts P (2009) Second-phase sampling designs for non-stationary spatial variables. Geoderma 153(1–2):205–216 CrossRefGoogle Scholar
  17. Dielman TE (2005) Least absolute value regression: recent contributions. J Stat Comput Simul 75(4):263–286 CrossRefGoogle Scholar
  18. Falivene O, Cabrera L, Tolosana-Delgado R, Sáez A (2010) Interpolation algorithm ranking using cross-validation and the role of smoothing effect. A coal zone example. Comput Geosci 36(4):512–519 CrossRefGoogle Scholar
  19. Fan J, Hall P (1994) On curve estimation by minimizing mean absolute deviation and its implications. Ann Stat 22(2):867–885 CrossRefGoogle Scholar
  20. Felicísimo AM (1994) Parametric statistical method for error detection in digital elevation models. ISPRS J Photogramm Remote Sens 49(4):29–33 CrossRefGoogle Scholar
  21. Fisher PF, Tate NJ (2006) Causes and consequences of error in digital elevation models. Prog Phys Geogr 30(4):467–489 CrossRefGoogle Scholar
  22. Franke R (1982) Scattered data interpolation: tests of some method. Math Comput 38(4):181–200 Google Scholar
  23. Franke R, Hagen H, Nielson G (1994) Least squares surface approximation to scattered data using multiquadratic functions. Adv Comput Math 2(1):81–99 CrossRefGoogle Scholar
  24. Golden Software I (2011) Surfer 10.3 users’ guide. Golden Software Inc., Golden, Colorado Google Scholar
  25. Gong J, Li Z, Zhu Q, Sui H, Zhou Y (2000) Effects of various factors on the accuracy of DEMs: an intensive experimental investigation. Photogramm Eng Remote Sens 66(9):1113–1117 Google Scholar
  26. Gonga-Saholiariliva N, Gunnell Y, Petit C, Mering C (2011) Techniques for quantifying the accuracy of gridded elevation models and for mapping uncertainty in digital terrain analysis. Prog Phys Geogr 35(6):739–764 CrossRefGoogle Scholar
  27. Goovaerts P (1997) Geostatistics for natural resources evaluation. Oxford University Press, New York, USA Google Scholar
  28. Hannah MJ (1981) Error detection and correction in digital terrain models. Photogramm Eng Remote Sens 47(1):63–69 Google Scholar
  29. Hardy RL (1971) Multiquadric equations of topography and other irregular surfaces. J Geophys Res 76(8):1905–1915 CrossRefGoogle Scholar
  30. Huang F, Liu D, Tan X, Wang J, Chen Y, He B (2011) Explorations of the implementation of a parallel IDW interpolation algorithm in a Linux cluster-based parallel GIS. Comput Geosci 37(4):426–434 CrossRefGoogle Scholar
  31. Huber P (1964) Robust estimation of a location parameter. Ann Math Stat 35(1):73–101 CrossRefGoogle Scholar
  32. Hutchinson MF (1989) A new procedure for gridding elevation and stream line data with automatic removal of spurious pits. J Hydrol 106(3–4):211–232 CrossRefGoogle Scholar
  33. Hutchinson MF, Gessler PE (1994) Splines—more than just a smooth interpolator. Geoderma 62(1–3):45–67 CrossRefGoogle Scholar
  34. Johnson ME, Moore LM, Ylvisaker D (1990) Minimax and maximin distance designs. J Stat Plan Inference 26(2):131–148 CrossRefGoogle Scholar
  35. Kiountouzis E (1973) Linear programming techniques in regression analysis. Applied Statistics, 69–73 Google Scholar
  36. Lam NSN (1983) Spatial interpolation methods: a review. Cartogr Geogr Inf Sci 10(2):129–150 CrossRefGoogle Scholar
  37. Li J, Chen C (2002) A simple efficient algorithm for interpolation between different grids in both 2D and 3D. Math Comput Simul 58(2):125–132 CrossRefGoogle Scholar
  38. Liu H, Jezek KC, O’Kelly ME (2001) Detecting outliers in irregularly distributed spatial data sets by locally adaptive and robust statistical analysis and GIS. Int J Geogr Inf Sci 15(8):721–741 CrossRefGoogle Scholar
  39. López C (1997) Locating some types of random errors in digital terrain models. Int J Geogr Inf Sci 11(7):677–698 CrossRefGoogle Scholar
  40. Makarovic B (1973) Progressive sampling for digital terrain models. ITC J 3:397–416 Google Scholar
  41. Maronna RA, Martin RD, Yohai VJ (2006) Robust statistics: theory and methods. Wiley Blackwell, New York CrossRefGoogle Scholar
  42. McMahon JR, Franke R (1992) Knot selection for least squares thin plate splines. SIAM J Sci Stat Comput 13:484–498 CrossRefGoogle Scholar
  43. Meer P, Mintz D, Rosenfeld A, Kim DY (1991) Robust regression methods for computer vision: a review. Int J Comput Vis 6(1):59–70 CrossRefGoogle Scholar
  44. Michalewicz Z, Fogel DB (2004) How to solve it: modern heuristics. Springer, Berlin CrossRefGoogle Scholar
  45. Mitas L, Mitasova H (1999) Spatial interpolation. In: Longley PA, Goodchild MF, Maguire MF, Rhind DJ (eds) Geographical information systems. Wiley, New York, pp 481–492 Google Scholar
  46. Mitasova H, Hofierka J (1993) Interpolation by regularized spline with tension: II. Application to terrain modeling and surface geometry analysis. Math Geol 25(6):657–669 CrossRefGoogle Scholar
  47. Müller WG (2007) Collecting spatial data: optimum design of experiments for random fields. Springer, Berlin Google Scholar
  48. Myers DE (1994) Spatial interpolation: an overview. Geoderma 62(1–3):17–28 CrossRefGoogle Scholar
  49. Nychka D, Yang Q, Royle JA (1997) Constructing spatial designs using regression subset selection. Stat Environ 3:131–154 Google Scholar
  50. Oliver MA, Webster R (1990) Kriging: a method of interpolation for geographical information systems. Int J Geogr Inf Sci 4(3):313–332 CrossRefGoogle Scholar
  51. Pardo-Igúzquiza E (1998) Optimal selection of number and location of rainfall gauges for areal rainfall estimation using geostatistics and simulated annealing. J Hydrol 210(1–4):206–220 CrossRefGoogle Scholar
  52. Portnoy S, Koenker R (1997) The Gaussian hare and the Laplacian tortoise: computability of squared-error versus absolute-error estimators. Stat Sci 12(4):279–300 CrossRefGoogle Scholar
  53. Rippa S (1999) An algorithm for selecting a good value for the parameter c in radial basis function interpolation. Adv Comput Math 11(2):193–210 CrossRefGoogle Scholar
  54. Rogerson PA, Delmelle E, Batta R, Akella M, Blatt A, Wilson G (2004) Optimal sampling design for variables with varying spatial importance. Geogr Anal 36(2):177–194 Google Scholar
  55. Rousseeuw PJ (1984) Least median of squares regression. J Amer Stat Assoc 79:871–880 CrossRefGoogle Scholar
  56. Royle JA, Nychka D (1998) An algorithm for the construction of spatial coverage designs with implementation in SPLUS. Comput Geosci 24(5):479–488 CrossRefGoogle Scholar
  57. Van Groenigen J, Pieters G, Stein A (2000) Optimizing spatial sampling for multivariate contamination in urban areas. Environmetrics 11(2):227–244 CrossRefGoogle Scholar
  58. Van Groenigen JW, Siderius W, Stein A (1999) Constrained optimisation of soil sampling for minimisation of the kriging variance. Geoderma 87(3–4):239–259 CrossRefGoogle Scholar
  59. Van Loan C (1985) On the method of weighting for equality-constrained least-squares problems. SIAM J Numer Anal 22(5):851–864 CrossRefGoogle Scholar
  60. Wagner HM (1959) Linear programming techniques for regression analysis. J Amer Stat Assoc 54:206–212 CrossRefGoogle Scholar
  61. Wang FT Scott DW (1994) The L1 method for robust nonparametric regression. J Amer Stat Assoc 89:65–76 Google Scholar
  62. Wang JF, Haining R, Cao ZD (2010) Sample surveying to estimate the mean of a heterogeneous surface: reducing the error variance through zoning. Int J Geogr Inf Sci 24(4):523–543 CrossRefGoogle Scholar
  63. Wang JF, Stein A, Gao BB, Ge Y (2012) A review of spatial sampling. Spatial Stat 2:1–14 Google Scholar
  64. Weibel R, Heller M (1993) Digital terrain modelling. In: Maguire DJ, Goodchild MF, Rhind DW (eds) Geographical information systems. Principles and applications. Oxford University Press, Longman, London, pp 269–297 Google Scholar

Copyright information

© International Association for Mathematical Geosciences 2013

Authors and Affiliations

  1. 1.Geomatics CollegeShandong University of Science and TechnologyQingdaoChina

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