Geophysical parameterization and interpolation of irregular data using natural neighbours

An approach is presented for interpolating a property of the Earth (e.g. temperature, or seismic velocity) specified at a series of `reference' points with arbitrary distribution in two or three dimensions. The method makes use of some powerful algorithms from the field of computational geometry to efficiently partition the medium into `Delaunay' triangles (in 2-D) or tetrahedra (in 3-D) constructed around the irregularly spaced reference points. The field can then be smoothly interpolated anywhere in the medium using a method known as natural neighbour interpolation. This method has the following useful properties:

We have extended the theory to produce expressions for the derivatives of the interpolated function. These may be calculated efficiently by modifying an existing algorithm which calculates the interpolated function using only local information. Full details of the theory and numerical algorithms are given. The new theory for function and derivative interpolation has applications to a range of geophysical interpolation and parameterization problems. In addition it shows much promise when used as the basis of a finite element procedure for numerical solution of partial differential equations.