Modelling three-dimensional geoscientific fields with the Voronoi diagram and its dual

Fields as found in the geosciences have properties that are not usually found in
other disciplines: the phenomena studied are often three-dimensional (3D), they
tend to change continuously over time, and the collection of samples to study the
phenomena is problematic, which often results in highly anisotropic distributions
of samples. In the geographical information system (GIS) community, raster
structures (voxels or octrees) are the most popular solutions, but, as we show in
this paper, they have shortcomings for modelling and analysing 3D geoscientific
fields. As an alternative to using rasters, we propose a new spatial model based
on the Voronoi diagram (VD) and its dual the Delaunay tetrahedralisation (DT),
and argue that they have many advantages over other tessellations. We discuss
the main properties of the 3D VD/DT, present some GIS operations that are
greatly simplified when the VD/DT is used, and, to analyse two or more fields, we
also present a variant of the map algebra framework where all the operations are
performed directly on VDs. The usefulness of this Voronoi-based spatial model is
demonstrated with a series of potential applications.