A high speed method of SMTS

In order to improve the computational speed of the method of surface modeling based on the theorem of surfaces (SMTS), a modified Gauss-Seidel (GS) method (MGS) was introduced and a MGS of SMTS (SMTS-MGS) has been developed. Numerical tests show that SMTS-MGS is more than twice as fast as SMTS-GS and much faster than the classical iterative methods provided by MATLAB 7.0. The computing time of SMTS-MGS is proportional to the first power of the total number of grid cells in the computational domain, whereas the classical SMTS computing time is proportional to the third power of the total number of grid cells. A real-world example of constructing a series of DEMs of Dongzhi tableland was employed to comparatively analyze the simulation accuracies of the two versions of SMTS including SMTS-MGS and the classical SMTS, and the well parameterized classical interpolation methods including the inverse distance weighted technique (IDW), kriging, thin plate spline (TPS), regularized TPS, TPS with tension, and ANUDEM Version 4.6.3, at spatial resolutions of 5, 12, 20, and 25m. The real-world example demonstrates that SMTS-MGS with the same accuracy as SMTS is approximately as accurate as the third-order TPS and much better than other classical interpolation methods at almost all spatial resolutions, except for ANUDEM at a spatial resolution of 5m. On average, SMTS-MGS provides the best results with a minimum of computing time.

[1]  M. Hutchinson.  A new procedure for gridding elevation and stream line data with automatic removal of spurious pits , 1989 .

[2]  David J. Maguire,et al.  Geographical Information Systems , 1993 .

[3]  Norbert Silvera,et al.  Accuracy of interpolation techniques for the derivation of digital elevation models in relation to landform types and data density , 2006 .

[4]  J. Carter,et al.  The Effect Of Data Precision On The Calculation Of Slope And Aspect Using Gridded Dems , 1992 .

[5]  Åke Björck,et al.  Iterative refinement of linear least squares solutions II , 1967 .

[6]  Timothy C. Coburn,et al.  Geostatistics for Natural Resources Evaluation , 2000, Technometrics.

[7]  Michael F. Hutchinson,et al.  Scale dependence in terrain analysis , 1997 .

[8]  Åke Björck,et al.  Numerical methods for least square problems , 1996 .

[9]  Mark P. Kumler.  An Intensive Comparison of Triangulated Irregular Networks (TINs) and Digital Elevation Models (DEMs) , 1994 .

[10]  Vladimir D. Liseikin,et al.  A Computational Differential Geometry Approach to Grid Generation (Scientific Computation) , 2003 .

[11]  M. Hutchinson,et al.  Splines — more than just a smooth interpolator , 1994 .

[12]  Gabriele Bitelli,et al.  Comparison of Techniques for Generating Digital Terrain Models from Contour Lines , 1997, Int. J. Geogr. Inf. Sci..

[13]  P. Goovaerts.  Geostatistical approaches for incorporating elevation into the spatial interpolation of rainfall , 2000 .

[14]  Philippe J. J. Desmet,et al.  Effects of Interpolation Errors on the Analysis of DEMs , 1997 .

[15]  Tianxiang Yue,et al.  A method of DEM construction and related error analysis , 2010, Comput. Geosci..

[16]  R. Webster,et al.  Kriging: a method of interpolation for geographical information systems , 1990, Int. J. Geogr. Inf. Sci..

[17]  C. Loan.  On the Method of Weighting for Equality Constrained Least Squares Problems , 1984 .

[18]  Chuanfa Chen,et al.  An application of Coons patch to generate grid-based digital elevation models , 2011, Int. J. Appl. Earth Obs. Geoinformation.

[19]  Michael F. Goodchild,et al.  Communicating uncertainty in spatial databases , 1996, Trans. GIS.

[20]  Nicholas C. Coops,et al.  Evaluating error associated with lidar-derived DEM interpolation , 2009, Comput. Geosci..

[21]  P. Burrough,et al.  Principles of geographical information systems , 1998 .

[22]  Tianxiang Yue,et al.  High-accuracy surface modelling and its application to DEM generation , 2010 .

[23]  Peter F. Fisher,et al.  Causes and consequences of error in digital elevation models , 2006 .

[24]  A. Gray,et al.  Modern Differential Geometry of Curves and Surfaces with Mathematica, Third Edition (Studies in Advanced Mathematics) , 2006 .

[25]  Nenad Ujevic.  A new iterative method for solving linear systems , 2006, Appl. Math. Comput..

[26]  Gil Gonçalves,et al.  Preserving cartographic quality in DTM interpolation from contour lines , 2002 .

[27]  F. Trochu.  A contouring program based on dual kriging interpolation , 1993, Engineering with Computers.

[28]  Saffet Erdogan,et al.  A comparision of interpolation methods for producing digital elevation models at the field scale , 2009 .

[29]  G. Wahba.  Spline models for observational data , 1990 .

[30]  J. N. Callow,et al.  How does modifying a DEM to reflect known hydrology affect subsequent terrain analysis , 2007 .

[31]  H. Mitásová,et al.  General variational approach to the interpolation problem , 1988 .

[32]  O. Dubrule.  Comparing splines and kriging , 1984 .

[33]  P. Kyriakidis,et al.  Error in a USGS 30-meter digital elevation model and its impact on terrain modeling , 2000 .

[34]  Stephen Wise,et al.  Assessing the quality for hydrological applications of digital elevation models derived from contours , 2000 .

[35]  Jay Gao,et al.  Resolution and Accuracy of Terrain Representation by Grid DEMs at a Micro-Scale , 1997, Int. J. Geogr. Inf. Sci..

[36]  Y. D. Huang.  Evaluation of Information Loss in Digital Elevation Models With Digital Photogrammetric Systems , 2000 .

[37]  V. Rovenski,et al.  Differential Geometry of Curves and Surfaces , 1952, Nature.

[38]  Kirsten M. de Beurs,et al.  Kriging and thin plate splines for mapping climate variables , 2001 .

[39]  Igor V. Florinsky,et al.  Accuracy of Local Topographic Variables Derived from Digital Elevation Models , 1998, Int. J. Geogr. Inf. Sci..

[40]  A. Kravchenko.  Influence of Spatial Structure on Accuracy of Interpolation Methods , 2003 .

[41]  Richard Franke,et al.  Thin plate splines with tension , 1985, Comput. Aided Geom. Des..

[42]  Ian Briggs.  Machine contouring using minimum curvature , 1974 .

[43]  G. Gilat,et al.  Method for smooth approximation of data , 1977 .

[44]  S. M. Wise,et al.  Effect of differing DEM creation methods on the results from a hydrological model , 2007, Comput. Geosci..

[45]  Tianxiang Yue,et al.  A new method of surface modeling and its application to DEM construction , 2007 .

[46]  PETER F. FISHER,et al.  Algorithm and Implementation Uncertainty in Viewshed Analysis , 1993, Int. J. Geogr. Inf. Sci..

[47]  Lubos Mitas,et al.  Multivariate Interpolation of Precipitation Using Regularized Spline with Tension , 2002, Trans. GIS.

[48]  Stuart N. Lane,et al.  Landform monitoring, modelling, and analysis , 1998 .

[49]  David W. S. Wong,et al.  An adaptive inverse-distance weighting spatial interpolation technique , 2008, Comput. Geosci..

[50]  Gerard B. M. Heuvelink,et al.  About regression-kriging: From equations to case studies , 2007, Comput. Geosci..