Two-dimensional interpolation of irregularly spaced data using polynomial splines

Abstract

A method has been developed for interpolating field data specified at irregularly spaced points over an area, such as is often the case with geophysical observations. The area is divided into triangles, of which the data points form the vertices. The interpolating function consists of a polynomial within each triangle. Across the boundary of each adjacent pair of triangles the values of the polynomials and their derivatives are matched to a prescribed degree. Third-degree polynomials permit matching of second derivatives, and fifth-degree polynomials are required for matching second derivatives. Boundary conditions may be imposed on the derivatives, and equations may be introduced to improve the smoothness of the interpolating function. The analytical nature of the function permits certain manipulations, for example calculation of directions of rays and intersection points in refraction seismology. The function could also be used for automatic contouring.