The Use of Structural-Parametric Approach for Approximation of Terrain Relief
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he Use of Structural-Parametric Approach for Approximation of Terrain Relief I. E. Stepanovaa, b, *, A. V. Shepetilovc, and V. V. Pogorelova, b aSchmidt
Institute of Physics of the Earth, Russian Academy of Sciences, Moscow, 123242 Russia bSirius University of Science and Technology, Sochi, 354340 Russia c Moscow State University, Department of Physics, Moscow, 119991 Russia *e-mail: [email protected] Received January 20, 2019; revised July 7, 2019; accepted July 8, 2020
Abstract—The structural–parametric approach is applied for constructing the analytical terrain models in the context of the method of modified S-approximations is considered. The results of the mathematical experiment with the use of the real data for terrains of hilly and mountainous topography are presented. Keywords: digital elevation models, approximation, relief, linear integral representations, systems of linear algebraic equations (SLAE), spatial S-approximations, structural-parametric method, block contrasting, Chebyshev’s three-layer iterative method DOI: 10.1134/S1069351320060105
INTRODUCTION Two versions of S-approximations—local and regional ones—are modifications of the method of linear integral representations. Main characteristics of the method are fairly thoroughly described in, e.g., (Strakhov and Stepanova, 2000; 2002a; 2002b; Stepanova, 2007; 2008; 2009; Stepanova et al., 2017; 2018a; 2018b; 2018c; Kerimov et al., 2018). To date, a great variety of methods and approaches has been developed for solving the interpretation problems of gravitational, magnetic, electrical, thermal and other fields of the Earth and planets (Balk and Dolgal, 2015a; 2015b; Dolgal et al., 2015; Muravina et al., 2019; Strakhov and Strakhov, 1999; Strakhov et al., 2009; Yagola et al., 2014; Alvarez et al., 2012; Anderson, 1976; Asgharzadeh, 2007; Barnes and Lumley, 2011; Barnes and Barraud, 2012; Bhattacharyya and Navolio, 2010; Blakely, 1995; Bouman et al., 2015; 2016; Braitenberg, 2015; Deng et al., 2016; Ditmar et al., 2003; Du et al., 2015). Terrain topography and potential fields are expanded in spherical harmonics, decomposed in the fields of the point sources, the recorded signals are Fourier analyzed, and all these representations need regularization algorithms since finding the coefficients for any Fourier series expansion is an ill-posed problem (Tikhonov et al., 1990). The modern digital elevation models are, most frequently, regular grids of cells of a given size or irregular triangular grids. Analytical topography models are another type of representation of the terrestrial or planetary surface data. In contrast to digital models with a piecewise continuous dependence of elevation
of the observation point on its projection coordinates, analytical models imply a more complex relationship between the elevation of the observation point above the surface and the geographical coordinates of the point. Simple models, e.g., spline quadrics, are widely used in the different geographic information systems (GIS) for approximating small segments of the Earth’s
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