Geophysical parametrization and interpolation of irregular data using natural neighbours

Abstract

S U M M A R YAn approach is presented for interpolating a property of the Earth (for example temperature or seismic velocity) specified at a series of 'reference' points with arbitrary distribution in two or three dimensions. The method makes use of some powerful algorithms from the field of computational geometry to efficiently partition the medium into 'Delaunay' triangles (in 2-D) or tetrahedra (in 3-D) constructed around the irregularly spaced reference points. The field can then be smoothly interpolated anywhere in the medium using a method known as natural-neighbour interpolation. This method has the following useful properties: (1) the original function values are recovered exactly at the reference points; (2) the interpolation is entirely local (every point is only influenced by its natural-neighbour nodes); and (3) the derivatives of the interpolated function are continuous everywhere except at the reference points. In addition, the ability to handle highly irregular distributions of nodes means that large variations in the scale-lengths of the interpolated function can be represented easily. These properties make the procedure ideally suited for 'gridding' of irregularly spaced geophysical data, or as the basis of parametrization in inverse problems such as seismic tomography.We have extended the theory to produce expressions for the derivatives of the interpolated function. These may be calculated efficiently by modifying an existing algorithm which calculates the interpolated function using only local information. Full details of the theory and numerical algorithms are given. The new theory for function and derivative interpolation has applications to a range of geophysical interpolation and parametrization problems. In addition, it shows much promise when used as the basis of a finite-element procedure for numerical solution of partial differential equations.

Updated Bouguer anomalies of the Iberian Peninsula: a new perspective to interpret the regional geology

“…In our database, the distance between points varies from 200 to c. 50,000 m. According to Sambridge et al (1995), the optimum grid spacing ranges between 100 and 25,000 m. If we used the former grid spacing, some artefacts would be created whereas using the latter would result in an appreciable loss of information. The adopted solution was to calculate the grid spacing from the weighted average distance between TIN nodes, which is 2000 m. See Ayala (2013) for a complete explanation of the procedure; further discussions about gridding can be found in previous work, for example, Rubio and Plata (1998); Smith and Wessel (1990); Sambridge et al (1995). The data are stored in both geographical coordinates (Longitude/Latitude; 10°W to 5°E, 35°50′ N to 44°N) and rectangular coordinates (UTM 30N in m; −75,000 to 113,600 easting, 3,980,000 to 4,871,000 northing), using the ETRS89 datum.…”

“…The optimum grid spacing has been calculated using Esri ArcMap. As the stations of the database have extremely inhomogeneous distribution, to grid the data we have followed the steps described in Sambridge, Braun, and McQueen (1995): (a) create a triangulated mesh (called TIN, Triangular Irregular Networks); (b) estimate the best grid spacing and (c) grid TIN using the Natural Neighbour algorithm. In our database, the distance between points varies from 200 to c. 50,000 m. According to Sambridge et al (1995), the optimum grid spacing ranges between 100 and 25,000 m. If we used the former grid spacing, some artefacts would be created whereas using the latter would result in an appreciable loss of information.…”

“…As the stations of the database have extremely inhomogeneous distribution, to grid the data we have followed the steps described in Sambridge, Braun, and McQueen (1995): (a) create a triangulated mesh (called TIN, Triangular Irregular Networks); (b) estimate the best grid spacing and (c) grid TIN using the Natural Neighbour algorithm. In our database, the distance between points varies from 200 to c. 50,000 m. According to Sambridge et al (1995), the optimum grid spacing ranges between 100 and 25,000 m. If we used the former grid spacing, some artefacts would be created whereas using the latter would result in an appreciable loss of information. The adopted solution was to calculate the grid spacing from the weighted average distance between TIN nodes, which is 2000 m. See Ayala (2013) for a complete explanation of the procedure; further discussions about gridding can be found in previous work, for example, Rubio and Plata (1998); Smith and Wessel (1990); Sambridge et al (1995).…”

Study of the high speed blanking proccess with the C-NEM

“…The NEM (Natural Element Method) interpolant is based on the Sibson's natural neighbor coordinates (shape functions) [3,10] and is constructed on the basis of the Voronoi diagram. For a set of nodes S = {n 1 , n 2 , .…”

Bayesian inversion of refraction seismic traveltime data

“…For the generation of the regular grid v(x, z), we first have to know in which triangle a particular gridpoint is located, and then to interpolate within this triangle. The efficient search of the encircling triangle is performed by the walking triangle or trifind algorithm (Lawson 1977;Lee & Schachter 1980;Sambridge et al 1995) without the need to check all triangles. The interpolation within a particular triangle is done by barycentric interpolation well known from computer graphics (e.g.…”