GEOPHYSICAL PARAMETRIZATION AND INTERPOLATION OF IRREGULAR DATA USINGNATURAL NEIGHBORS

An approach is presented for interpolating a property of the Earth (fo r example temperature or seismic velocity) specified at a series of 'r eference' points with arbitrary distribution in two or three dimension s. The method makes use of some powerful algorithms from the field of computational geometry to efficiently partition the medium into 'Delau nay' triangles (in 2-D) or tetrahedra (in 3-D) constructed around the irregularly spaced reference points. The field can then be smoothly in terpolated anywhere in the medium using a method known as natural-neig hbour interpolation. This method has the following useful properties: (1) the original function values are recovered exactly at the referenc e points; (2) the interpolation is entirely local (every point is only influenced by its natural-neighbour nodes); and (3) the derivatives o f the interpolated function are continuous everywhere except at the re ference points. In addition, the ability to handle highly irregular di stributions of nodes means that large variations in the scale-lengths of the interpolated function can be represented easily. These properti es make the procedure ideally suited for 'gridding' of irregularly spa ced geophysical data, or as the basis of parametrization in inverse pr oblems such as seismic tomography. We have extended the theory to prod uce expressions for the derivatives of the interpolated function. Thes e may be calculated efficiently by modifying an existing algorithm whi ch 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 parametrization problems. In addition, it shows much promise when used as the basis of a finite-ele ment procedure for numerical solution of partial differential equation s.