An adaptive application of multiquadric interpolants for numerically modeling large numbers of irregularly spaced hydrographic data

Abstract

The approximation or interpolation of irregularly spaced data in two or higher dimensions is extremely difficult in contrast to the one dimensional case. No one method works well in all instances with the possible exception of least squares approximation. One is faced with choosing between many techniques, each having been applied to only a relatively small amount of data, whether a local or global approximation or interpolation is desired. In a study of many different methods, each applied to a variety of test surfaces, multiquadric interpolants were shown to be some of the best global methods tested. This dissertation explores one technique for applying multiquadric interpolants to large numbers of irregularly spaced two dimensional data. The technique involves building a data structure to partition the domain of the data into a set of smaller divisions called cells. Any adjacent cells whose data are similar are then adaptively combined into one group. After grouping of the cells, multiquadric interpolants are then iteratively applied to each group of cells until the maximum error between the resulting data model and all original data are within a user specified tolerance. This gives a model for each cell or group of cells which locally approximates in an l(,(INFIN)) sense or locally interpolates the original data. A method for joining group models across group boundaries is also explored in this dissertation. A set of weighting functions was employed such that the models joined smoothly to the specified degree of continuity. In the test cases, the approximation or interpolation property of the original data was preserved. The method was run on five representative sets of hydrographic data, consisting of over 12,000 points, and two sets of data computed from common, mathematically defined, surfaces in the literature. For the largest set of data which was comprised of over 3000 points, the partitioning and grouping required 28 seconds, the modeling required 11.5 minutes, and the computation of a regular grid from the model consisting of 20,000 points required 3 minutes of AMDAHL 470 V/6 CPU time.