Numerical Integration on the Sphere

Abstract

This chapter is concerned with numerical integration over the unit sphere $${\mathbb{S}}^{2} \subset {\mathbb{R}}^{3}$$. We first discuss basic facts about numerical integration rules with positive weights. Then some important types of rules are discussed in detail: rules with a specified polynomial degree of precision, including the important case of longitude--latitude rules; rules using scattered data points; rules based on equal-area partitions; and rules for numerical integration over subsets of the sphere. Finally we show that for numerical integration over the whole sphere and for functions with an appropriate degree of smoothness, an optimal rate of convergence can be achieved by positive-weight rules with polynomial precision, and also by rules obtained by integrating a suitable radial basis function interpolant.