Interpolation by Regularized Spline with Tension : I. Theory and Implementation

Bivariate and trivariate functions for interpolation from scattered data are derived. They are constructed by explicit minimization of a general smoothness functional, and they include a tension parameter that controls the character of the interpolation function (e.g., for bivariate case the surface can be tuned from a `membrane' to a `thin steel plate'). Tension can be applied also in a chosen direction, for modeling of phenomena with a simple type of anisotropy. The functions have regular derivatives of all orders everywhere. This makes them suitable for analysis of surface geometry and for direct application in models where derivatives are necessary. For processing of large data sets (thousands of data points), which are now common in geosciences, a segmentation algorithm with a exible size of overlapping neighborhood is presented. Simple examples demonstrating exibility and accuracy of the functions are presented.