Spatial Interpolation: From Two to Three Dimensions

Interpolation methods are an important part in a geographical information system (GIS) and have been used for years to model elevation data. They are crucial in the visualisation process (generation of contours), for the conversion of data from one format to another, to have a better understanding of a dataset or simply to identify bad samples. The result of interpolation—usually a surface that represents the real terrain—must be as accurate as possible because it often forms the basis for spatial analysis, e.g. runoff modelling or visibility analysis. Although interpolation helps in creating three-dimensional surfaces, it is intrinsically a two-dimensional operation for only the x-y coordinates of each sample are used and the elevation is considered as an attribute. With the new technologies available to collect information about the Earth, more and more three-dimensional data are collected. A typical dataset in geosciences has samples in 3D space (x-y-z coordinates) to which attributes are attached (e.g. the salinity of the water, or the percentage of gold in the rock). Because of the way they are collected, 3D geoscientific datasets have a highly anisotropic distribution. Geologic and oceanographic data are for example respectively gathered from boreholes and water columns; data are therefore usually abundant vertically but sparse horizontally. While most of the interpolation methods used in GIS intuitively extend to 3D, it is not obvious that they preserve their properties or are appropriate for such datasets. In this paper, we discuss the extension to 3D of some of the interpolation methods found in GIS or geoscientific modelling packages. We first present some details concerning their generalisation, and then evaluate briefly their properties.