MODELLING OF 3 DIMENSIONAL SCALAR FIELDS REPRESENTED BY FUNCTION VALUES IN SCATTERED POINTS

Abstract

The phenomena of the real world exist in three dimensional space. The authors in the former years performed a research aiming at modelling the undersurface solids, cavities and strata. The present paper is the first publication concerning the continuation of this research. In this new activity, an attempt will be made to model scalar fields of different properties under and over the surface of the Earth. The modelling process is influenced by the character of the source data which depends on the conditions of the data acquisition. Especially in undersurface models - it is very expensive to capture new data; - there is no possibility to measure in the places of maximum and minimum; - the values of arguments as well as of functions are determined with several types of uncertainities. The methods to be applied should take into account the properties of the source data as well as the problems to be solved by the model. After sketching the methods to be examined in the future, the method of local polynomials worked out by the authors is explained in detail. The gist of this approach is to divide the set of nodal points into M "units", composed by help of Voronoi polyhedrons. Each unit represents a polynomial of k-th degree (where k depends on the accuracy of the source data). The interpolation of a function value for a point P in the competency of the i-th unit's polynomial can be calculated as a weighted average of all the M polynomials for the argument xp, yp, zp. The weight is growing when the distance of the unit is decreasing.