Mathematical Geosciences

, Volume 46, Issue 8, pp 909–929 | Cite as

Smooth Surface Modeling of DEMs Based on a Regularized Least Squares Method of Thin Plate Spline



Thin plate spline (TPS) has been widely accepted as a method for smooth fitting of noisy data. However, the classical TPS always has an ill-conditioning problem when two sample points are very close. Although the modified orthogonal least squares-based TPS (TPS-M) avoids this ill-conditioning problem, it is not completely immune to over-fitting when sample points are noisy. In this paper, a regularized least squares method of thin plate spline (TPS-RLS) was developed, which adds a weight penalty term to the error criterion of orthogonal least squares (OLS). TPS-RLS combines the advantages of both regularization and OLS, which avoid the over-fitting and the ill-conditioning problems simultaneously. Numerical tests indicate that irrespective of the standard deviation of sampling errors and the number of knots, TPS-RLS is always more accurate than TPS-M for smooth fitting of noisy data, whereas TPS-M would have a serious over-fitting problem if the optimal number of knots were not determined in advance. The real-world example of fitting total station instrument data shows that among the classical interpolation methods including IDW, natural neighbor and ordinary kriging, TPS-RLS has the highest accuracy for a series of DEMs with different resolutions, especially for the coarse one. Surface modeling of DEMs with contour lines demonstrate that TPS-RLS has a better performance than the classical methods in terms of both root mean squared error and relief shaded map appearance.


DEM Thin plate spline Noise Regularization 



Special thanks go to the two anonymous reviewers for their assistance, comments and suggestions. This work was supported by the National Natural Science Foundation of China (Grant No. 41101433, 41371367), Young and Middle-Aged Scientists Research Awards Fund of Shandong Province (Grant No. BS2012HZ010), Qingdao Science and Technology Program of Basic Research Project (Grant No. 13-1-4-239-jch), the Key Laboratory of Marine Surveying and Mapping in Universities of Shandong (Shandong University of Science and Technology) (Grant No. 2013B03), and Special Project Fund of Taishan Scholars of Shandong Province.


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Copyright information

© International Association for Mathematical Geosciences 2014

Authors and Affiliations

  • Chuanfa Chen
    • 1
  • Yanyan Li
    • 1
  • Xuewei Cao
    • 1
  • Honglei Dai
    • 1
  1. 1.Geomatics CollegeShandong University of Science and TechnologyQingdaoChina

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