Scattered Data Fitting on the Sphere

We discuss several approaches to the problem of interpolating or approximating data given at scattered points lying on the surface of the sphere. These include methods based on spherical harmonics, tensor-product spaces on a rectangular map of the sphere, functions deened over spherical triangulations, spherical splines, spherical radial basis functions, and some associated multi-resolution methods. In addition, we brieey discuss sphere-like surfaces, visualization, and methods for more general surfaces. The paper includes a total of 206 references. x1. Introduction Let S be the unit sphere in IR 3 , and suppose that fv i g n i=1 is a set of scattered points lying on S. In this paper we are interested in the following problem: Problem 1. Given real numbers fr i g n i=1 , nd a (smooth) function s deened on S which interpolates the data in the sense that or approximates it in the sense that Data tting problems where the underlying domain is the sphere arise in many areas, including e.g. geophysics and meteorology where the sphere is taken as a model of the earth. The question of whether interpolation or approximation should be carried out depends on the setting, although in practice measured data are almost always noisy, in which case approximation is probably more appropriate. In most applications, we will want s to be at least continuous. In some cases we may want it to be C 1 so that the associated surface F := fs(v)v : v 2 Sg is tangent plane continuous. All rights of reproduction in any form reserved.