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Isarithmic Mapping
Published in Terry A. Slocum, Robert B. McMaster, Fritz C. Kessler, Hugh H. Howard, Thematic Cartography and Geovisualization, 2022
Terry A. Slocum, Robert B. McMaster, Fritz C. Kessler, Hugh H. Howard
Thin-plate splines involve fitting a mathematical surface to the data such that the roughness of the surface is minimized. Depending on the options chosen for interpolation, the surface will either pass directly through the values for control points or be close to those values. The term thin-plate is suggestive of how you should attempt to visualize the process—imagine bending a thin plate of flexible material (such as rubber) and trying to distort it to fit the values of the control points in three dimensions. Note that this approach is quite different from the weighted average of surrounding control points used in inverse distance and kriging. Since surrounding points are not averaged, it is possible for the resulting surface to extend beyond the limits of the input data (i.e., to predict values outside the range of the data). We were especially interested in experimenting with thin-plate splines because the technique is commonly utilized with climatic data (e.g., Tatalovich et al. (2006), Snehmani et al. (2015), Plouffe et al. (2015)).
Interpolation and Approximation Theory
Published in A. C. Faul, A Concise Introduction to Numerical Analysis, 2018
The two-dimensional equivalent of the cubic spline is called thin-plate spline. a linear combination of thin-plate splines passes through the data points exactly while minimizing the so-called bending energy, which is defined as the integral over the squares of the second derivatives ∬(fxx2+2fxy2+fyy2)dxdy
Thin plate spline Galerkin scheme for numerically solving nonlinear weakly singular Fredholm integral equations
Published in Applicable Analysis, 2019
In this paper, we have presented a simple and numerically interesting method for solving non-linear weakly singular integral equations of the second kind using the thin-plate splines in the discrete Galerkin method. We can regard the thin-plate splines as a type of the free shape parameter radial basis functions which are an effective technique for estimating an unknown function. The singular integrals occurred in this method have been computed utilizing a composite non-uniform Gauss-Legendre quadrature scheme. This approach does not require any background approximation cells and so it is a meshless method. We have also obtained the error bound for the scheme and found that the convergence rate of the proposed method is of . As demonstrated by the computational results reported in Section 5, the proposed technique is able to produce accurate solutions justified by the theoretical error estimates.
Quantitative risk evaluation for late rice: hazard factors in Zhejiang province, China
Published in Geomatics, Natural Hazards and Risk, 2020
Ran Cheng, Qin’ou Liang, Degen Lin
Spatial interpolation is the process of estimating the values of other points by using the data of known points. It is a method of converting point data into planar data (Chang 2010). The Anusplin interpolation model is a kind of spatial interpolation method, which developed by the Australian National University using the Fortran language based on the thin-plate smoothing splines. The thin-plate splines generates a surface that passes through the control points, and minimizes the slope change of all slopes formed by all the points, that is, the surface that generates the minimum curvature to fit the control points (Bookstein 1989; Hutchinson 1995). And the thin-plate smoothing splines function is an extension based on it. The function expression of the thin disk smooth splines function is shown in Eq. (9). where is a spline independent variable vector; yi is an independent covariate vector; is a random error.; is the dependent variable at point i of the space. Function and coefficient b are calculated by Eq. (10). where is weight; is called the smoothing parameter; The function is used to represent the roughness of the function is the number of splines in the Anusplin interpolation model (Merz et al. 2004; de Moel and Aerts 2011).