Abstract
A method of surface modeling, high accuracy surface modeling (HASM), which is based on the fundamental theorem of surface theory, is modified. The earlier version of HASM is theoretically incomplete and almost performs similar or slightly better than other methods being compared in the practical applications which definitely limit its promotion. According to the fundamental theorem of surface theory, we modify HASM by adding another important nonlinear equation to solve the low accuracy in some cases and make HASM have a complete and solid theory foundation. A numerical test and a real-world example are employed to comparatively validate the effectiveness of this modification. It is found that the accuracy of the simulation result has a great improvement. Another feature of the modified version of HASM is that it is theoretically perfect since it considers the third equation of the surface theory. The modified HASM will be useful with a wide range of spatial interpolation, particularly if the focus on simulation accuracy.
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Apaydin, H., Anli, A.S., Ozturk, F.: Evaluation of topographical and geographical effects on some climatic parameters in the Central Anatolis Region of Turkey. Int. J. Climatol. 31, 1264–1279 (2011)
Bjorck, A.: Iterative refinement of linear least squares solutions II. BIT 8, 8–30 (1968)
Brezinski, C., Rodriguez, G., Seatzu, S.: Error estimates for linear systems with applications to regularization. Numer. Algorithm 49, 85–104 (2008)
Carmo, M.P.: Differential Geometry of Curves and Surfaces. China Machine Press, Beijing (2006)
Chen, C.F., Yue, T.X.: A method of DEM construction and related error analysis. Comput. Geosci. 36, 717–725 (2010)
Chen, C.F., Yue, T.X., Li, Y.Y.: A high speed method of SMTS. Comput. Geosci. 41, 64–71 (2012)
Crain, I.K.: Digital representation of topographic surface. Photogramm Eng. Remote Sens 54, 1577 (1970)
Golub, G.H., Van Loan, C.F.: Matrix Computation, Johns Hopkins Series in the Mathematical Sciences, 3rd edn. Johns Hopkins University Press, Baltimore (1989)
Golub, G.H., Van Loan, C.F.: Matrix Computations. Posts & Telecom Press, Beijing (2009)
Goovaerts, P.: Geostatistics in soil science: state-of-the-art and perspectives. Geoderma 89, 1–45 (1999)
Hancock, P.A., Hutchinson, M.F.: Spatial interpolation of large climate data sets using bivariate thin plate smoothing splines. Environ. Model. Softw. 21, 1684–1694 (2006)
Hartkamp, A.D., De Beurs, K., Stein, A., White, J.W.: Interpolation techniques for climate variables. NRG-GIS Series 99-01, CIMMYT, Mexio (1999)
Henderson, D.W.: Differential Geometry. Prentice-Hall, London (1998)
Joly, D., Brossard, T., Cardot, H., Cavailhes, J., Hilal, M., Wavresky, P.: Temperature interpolation based on local information: the example of France. Int. J. Climatol. 31, 2141–2153 (2011)
Karniadakis, G.E.M., Kirby, I.I.R.M.: Parallel Scientific Computing in C++ and MPI. Cambridge University Press, Cambridge (2003)
Kurtzman, D., Kadmon, R.: Mapping of temperature variables in Isreal: a comparison of different interpolation methods. Clim. Res. 13, 33–43 (1999)
Li, Z.L., Zhu, Q.: Digital Elevation Model. Wuhan Technical University of Surveying and Mapping Press, Wuhan (2000)
Liao, S.B., Li, Z.H.: Some practical problems related to raserization of air temperature. Meteorol. Sci. Technol. 32, 352–356 (2004) (in Chinese)
Liseikin, V.D.: A Computational Differential Geometry Approach to Grid Generation. Springer, Berlin (2004)
Naumova, V., Pereverzyev, S.V., Sivananthan, S.: Adaptive parameter choice for one-side finite difference schemes and its application in diabetes technology. J. Complexity 28, 524–538 (2012)
Ninyerola, M., Pons, X., Roure, J.M.: Objective air temperature mapping for the Iberian Peninsula using spatial interpolation and GIS. Int. J. Climatol. 27, 1231–1242 (2007)
Reichel, L., Rodriguez, G., Seatzu, S.: Error estimates for large-scale ill-posed problems. Numer. Algorithms 51, 341–361 (2009)
Samanta, S., Pal, D.K., Lohar, D.: Interpolation of climate variables and temperature modeling. Theor. Appl. Climatol. 107, 35–45 (2012)
Somasundaram, D.: Differential Geometry. Alpha Science International Ltd, Harrow (2005)
Stott, J.P.: Surface Modeling by Computer. Thomas Telford Ltd for the Institution of Civil Engineers, London (1977)
Su, B.Q., Hu, H.S.: Differential Geometry. People’s Education Press, Beijing (1997). (in Chinese)
Toponogov, V.A.: Differential Geometry of Curves and Surfaces. Birkhaeuser Boston, New York (2006)
Wang, F.: Quantitative Methods and Applications in GIS. CRC Press, Boca Raton (2006)
Wise, S.: GIS data modeling-lessons from the analysis of DTMs. Int. J. Geogr. Inf. Sci. 14, 313–318 (2000)
Yue, T.X.: Surface Modeling: High Accuracy and High Speed Methods. CRC Press, New York (2011)
Yue, T.X., Du, Z.P.: Numerical test for optimum formulation of high accuracy surface modeling. Geo Inf. Sci. 8, 83–87 (2006) (in Chinese)
Yue, T.X., Wang, S.H.: Adjustment computation of HASM: a high-accuracy and high-speed method. Int. J. Geogr. Inf. Sci. 24, 1725–1743 (2010)
Yue, T.X., Du, Z.P., Song, D.J., Gong, Y.: A new method of high accuracy surface modeling and its application to DEM construction. Geomorphology 91, 161–172 (2007)
Yue, T.X., Chen, C.F., Li, B.L.: An adaptive method of high accuracy surface modeling and its application to simulating elevation surface. Trans. GIS 14, 615–630 (2010a)
Yue, T.X., Song, D.J., Du, Z.P., Wang, W.: High accuracy surface modeling and its application to DEM generation. Int. J. Remote Sens. 31, 2205–2226 (2010b)
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Zhao, N., Yue, T. & Zhao, M. An improved version of a high accuracy surface modeling method. Int J Geomath 4, 185–200 (2013). https://doi.org/10.1007/s13137-013-0051-z
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DOI: https://doi.org/10.1007/s13137-013-0051-z