Multivariate interpolation | EPFL Graph Search

In numerical analysis, multivariate interpolation is interpolation on functions of more than one variable (multivariate functions); when the variates are spatial coordinates, it is also known as spatial interpolation. The function to be interpolated is known at given points and the interpolation problem consists of yielding values at arbitrary points . Multivariate interpolation is particularly important in geostatistics, where it is used to create a digital elevation model from a set of points on the Earth's surface (for example, spot heights in a topographic survey or depths in a hydrographic survey). For function values known on a regular grid (having predetermined, not necessarily uniform, spacing), the following methods are available. Nearest-neighbor interpolation n-linear interpolation (see bi- and trilinear interpolation and multilinear polynomial) n-cubic interpolation (see bi- and tricubic interpolation) Kriging Inverse distance weighting Natural neighbor interpolation Spline interpolation Radial basis function interpolation Barnes interpolation Bilinear interpolation Bicubic interpolation Bézier surface Lanczos resampling Delaunay triangulation Bitmap resampling is the application of 2D multivariate interpolation in . Three of the methods applied on the same dataset, from 25 values located at the black dots. The colours represent the interpolated values. See also Padua points, for polynomial interpolation in two variables. Trilinear interpolation Tricubic interpolation See also bitmap resampling. Catmull-Rom splines can be easily generalized to any number of dimensions. The cubic Hermite spline article will remind you that for some 4-vector which is a function of x alone, where is the value at of the function to be interpolated. Rewrite this approximation as This formula can be directly generalized to N dimensions: Note that similar generalizations can be made for other types of spline interpolations, including Hermite splines.

Spatial variability of potassium in agricultural soils of the canton of Fribourg, Switzerland

Stéphane Joost, Nicolas Rossier, Guillaume Blanchet, André Schneider

Potassium (K) is a crucial element for plant nutrition and its availability and spatial distribution in agricultural soils is influenced by many agro-environmental factors. In Switzerland, a soil monitoring network (FRIBO) was established in 1987 with 250 sites distributed over the whole of the canton of Fribourg (representing 4% of the surface area of Switzerland), whose territory is shared between the Swiss Midlands and the Western Alp foothills. In this study area, diverse geological deposits (sandstone, marlstone, silts and calcareous rocks), soil types (Cambisols, Gleysols, Rendzinas, Luvisols and Fluvisols) and land uses (cropland, permanent grassland and mountain pasture) are present, making the network interesting for assessing the relative contribution of environmental variables and land use management on soil properties. The aims of the present study were to (i) characterize the soil K status in the Fribourg canton according to four different extraction methods; (ii) analyse the spatial variability of soil K in relation to land use, soil type, soil parent material and topography; (iii) evaluate the spatial predictability of K at the canton level; and (iv) analyse the implications for K fertilization management. The overall amount of soil total K averaged 13.6 g.kg-1 with significant variations across the sites (5.1-22.1 g.kg-1). The spatial distribution of total K was particularly influenced by soil parent materials, as suggested by a significant global spatial autocorrelation measure (Moran’s I10km = 0.43) and significant differences observed among soil types and soil parent materials. On the other hand, available mean K forms were significantly different among land uses, with the highest mean values of available K encountered in permanent grasslands, from 46.3 mg.kg-1 (water extraction) to 198 mg.kg-1 (acetate ammonium + EDTA extraction). All K forms showed similar spatial regional patterns for all spatial interpolation methods, with areas dominated by permanent grassland and crops presenting higher values. However, these trends were less pronounced for the available K forms due to the prevalence of on-farm management practices for these K forms and their high temporal variability. This hypothesis was supported by spatial clustering of low and/or high K fertility status that could be related to local particular farming practices. Grasslands require particular attention with regard to overall high K fertility status.

Elsevier2017

Spatial variability of potassium in agricultural soils of the canton of Fribourg, Switzerland

Stéphane Joost, Nicolas Rossier, Guillaume Blanchet, André Schneider

Elsevier2017