Algorithm for the Production of Contour Maps from Scattered Data

THE production of contour maps from data measured at a set of points randomly scattered over some given region is a common problem. For example, geologists are interested in the production of trend surfaces and the associated contour maps from data consisting of the x,y map coordinates of bore locations and the corresponding depths below sea level at which a certain type of rock was found. Although the points at which data are available are not random, in general they are not at points of a rectangular grid. Similarly, for meteorological purposes, although the sites for weather stations can be chosen more rationally they are neither sufficiently dense nor precisely sited to form a complete grid. Many techniques have been used to produce grid values from the given data, most of which use either some triangulation method or else a local or global fit of polynomial or Fourier surfaces. James1 indicates some of the disadvantages of these methods, in particular the fact that in regression of either polynomial or two-dimensional Fourier series both represent finite expansions of infinite series with an increasing number of extrema and inflexions on further expansion. Both are linear with respect to their coefficients so they are both specific cases of the general linear model and thus are subject to the mathematical restrictions inherent in such models. James and Krumbein2 list those restrictions of importance geologically. James describes a method whereby a non-linear model in general mathematical terms is assigned as an educated guess based on the user's geological intuition. The method is iterative in the sense that the original model can be revised after further analysis of the residuals.