Local numerical integration on the sphere

Many applications in geomathematics as well as bio-medical applications require the analysis of an unknown target function of a large amount of data, which can be modeled as data on a subset of the surface of a sphere. An important ingredient of this analysis is to develop numerical integration schemes (quadrature formulas) to integrate spherical polynomials of as high a degree as possible exactly. Since many subsets of the sphere can be subdivided efficiently into spherical triangles, the problem reduces to computing quadrature formulas for integration on spherical triangles. In this paper, we present an algorithm for computing quadrature formulas based on “scattered data” on a triangle; i.e., without requiring a theoretically prescribed choice of the location of these points. We present several numerical examples to illustrate various features of our algorithm in the context of both integration and function approximation.