Global spherical harmonic analysis by least-squares and numerical quadrature methods in historical perspective

Methods of global spherical harmonic analysis of discrete data on a sphere are placed in a historical context. The paper concentrates on the loss of orthogonality in the direction of latitude, due to the transition from continuous to discretized functions. Special attention is paid to Neumann's (1838) solution to this problem. By recasting the formulae of spherical harmonic analysis into matrix-vector notation, both least-squares solutions and quadrature methods are represented in a general framework of weighted least squares. It is also shown that the two-step formulation of global spherical harmonic computation was applied already by Neumann (1838) and Gauss (1839). Computational modifications to Neumann's method are reviewed as well.